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Theorem cantnfresb 43337
Description: A Cantor normal form which sums to less than a certain power has only zeros for larger components. (Contributed by RP, 3-Feb-2025.)
Assertion
Ref Expression
cantnfresb (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶) ↔ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹

Proof of Theorem cantnfresb
Dummy variables 𝑎 𝑏 𝑐 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . . . . . . . 11 dom (𝐴 CNF 𝐵) = dom (𝐴 CNF 𝐵)
2 eldifi 4131 . . . . . . . . . . . 12 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
32adantr 480 . . . . . . . . . . 11 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
4 simpr 484 . . . . . . . . . . 11 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
5 eqid 2737 . . . . . . . . . . 11 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)))} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)))}
61, 3, 4, 5cantnf 9733 . . . . . . . . . 10 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝐴 CNF 𝐵) Isom {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)))}, E (dom (𝐴 CNF 𝐵), (𝐴o 𝐵)))
76adantr 480 . . . . . . . . 9 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵)) → (𝐴 CNF 𝐵) Isom {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)))}, E (dom (𝐴 CNF 𝐵), (𝐴o 𝐵)))
8 simpr 484 . . . . . . . . 9 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵)) → 𝐹 ∈ dom (𝐴 CNF 𝐵))
9 ondif2 8540 . . . . . . . . . . . . . . 15 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
109simprbi 496 . . . . . . . . . . . . . 14 (𝐴 ∈ (On ∖ 2o) → 1o𝐴)
11 dif20el 8543 . . . . . . . . . . . . . 14 (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
1210, 11ifcld 4572 . . . . . . . . . . . . 13 (𝐴 ∈ (On ∖ 2o) → if(𝑦 = 𝐶, 1o, ∅) ∈ 𝐴)
1312ad2antrr 726 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝑦𝐵) → if(𝑦 = 𝐶, 1o, ∅) ∈ 𝐴)
1413fmpttd 7135 . . . . . . . . . . 11 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)):𝐵𝐴)
1511adantr 480 . . . . . . . . . . . 12 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → ∅ ∈ 𝐴)
16 eqid 2737 . . . . . . . . . . . 12 (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))
174, 15, 16sniffsupp 9440 . . . . . . . . . . 11 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) finSupp ∅)
181, 3, 4cantnfs 9706 . . . . . . . . . . 11 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) ∈ dom (𝐴 CNF 𝐵) ↔ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)):𝐵𝐴 ∧ (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) finSupp ∅)))
1914, 17, 18mpbir2and 713 . . . . . . . . . 10 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) ∈ dom (𝐴 CNF 𝐵))
2019adantr 480 . . . . . . . . 9 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵)) → (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) ∈ dom (𝐴 CNF 𝐵))
21 isorel 7346 . . . . . . . . 9 (((𝐴 CNF 𝐵) Isom {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)))}, E (dom (𝐴 CNF 𝐵), (𝐴o 𝐵)) ∧ (𝐹 ∈ dom (𝐴 CNF 𝐵) ∧ (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) ∈ dom (𝐴 CNF 𝐵))) → (𝐹{⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)))} (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) ↔ ((𝐴 CNF 𝐵)‘𝐹) E ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)))))
227, 8, 20, 21syl12anc 837 . . . . . . . 8 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵)) → (𝐹{⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)))} (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) ↔ ((𝐴 CNF 𝐵)‘𝐹) E ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)))))
2322adantrl 716 . . . . . . 7 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (𝐹{⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)))} (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) ↔ ((𝐴 CNF 𝐵)‘𝐹) E ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)))))
2423adantr 480 . . . . . 6 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ 𝐶𝐵) → (𝐹{⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)))} (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) ↔ ((𝐴 CNF 𝐵)‘𝐹) E ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)))))
25 fvexd 6921 . . . . . . 7 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ 𝐶𝐵) → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))) ∈ V)
26 epelg 5585 . . . . . . 7 (((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))) ∈ V → (((𝐴 CNF 𝐵)‘𝐹) E ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))) ↔ ((𝐴 CNF 𝐵)‘𝐹) ∈ ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)))))
2725, 26syl 17 . . . . . 6 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ 𝐶𝐵) → (((𝐴 CNF 𝐵)‘𝐹) E ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))) ↔ ((𝐴 CNF 𝐵)‘𝐹) ∈ ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)))))
282ad2antrr 726 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶𝐵) → 𝐴 ∈ On)
29 simplr 769 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶𝐵) → 𝐵 ∈ On)
30 fconst6g 6797 . . . . . . . . . . . . . . . . . 18 (∅ ∈ 𝐴 → (𝐵 × {∅}):𝐵𝐴)
3111, 30syl 17 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ (On ∖ 2o) → (𝐵 × {∅}):𝐵𝐴)
3231adantr 480 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝐵 × {∅}):𝐵𝐴)
334, 15fczfsuppd 9426 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝐵 × {∅}) finSupp ∅)
341, 3, 4cantnfs 9706 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → ((𝐵 × {∅}) ∈ dom (𝐴 CNF 𝐵) ↔ ((𝐵 × {∅}):𝐵𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
3532, 33, 34mpbir2and 713 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝐵 × {∅}) ∈ dom (𝐴 CNF 𝐵))
3635adantr 480 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶𝐵) → (𝐵 × {∅}) ∈ dom (𝐴 CNF 𝐵))
37 simpr 484 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶𝐵) → 𝐶𝐵)
3810ad2antrr 726 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶𝐵) → 1o𝐴)
39 fczsupp0 8218 . . . . . . . . . . . . . . . 16 ((𝐵 × {∅}) supp ∅) = ∅
40 0ss 4400 . . . . . . . . . . . . . . . 16 ∅ ⊆ 𝐶
4139, 40eqsstri 4030 . . . . . . . . . . . . . . 15 ((𝐵 × {∅}) supp ∅) ⊆ 𝐶
4241a1i 11 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶𝐵) → ((𝐵 × {∅}) supp ∅) ⊆ 𝐶)
43 0ex 5307 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
4443fvconst2 7224 . . . . . . . . . . . . . . . . 17 (𝑦𝐵 → ((𝐵 × {∅})‘𝑦) = ∅)
4544ifeq2d 4546 . . . . . . . . . . . . . . . 16 (𝑦𝐵 → if(𝑦 = 𝐶, 1o, ((𝐵 × {∅})‘𝑦)) = if(𝑦 = 𝐶, 1o, ∅))
4645mpteq2ia 5245 . . . . . . . . . . . . . . 15 (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ((𝐵 × {∅})‘𝑦))) = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))
4746eqcomi 2746 . . . . . . . . . . . . . 14 (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ((𝐵 × {∅})‘𝑦)))
481, 28, 29, 36, 37, 38, 42, 47cantnfp1 9721 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶𝐵) → ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) ∈ dom (𝐴 CNF 𝐵) ∧ ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))) = (((𝐴o 𝐶) ·o 1o) +o ((𝐴 CNF 𝐵)‘(𝐵 × {∅})))))
4948simprd 495 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶𝐵) → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))) = (((𝐴o 𝐶) ·o 1o) +o ((𝐴 CNF 𝐵)‘(𝐵 × {∅}))))
5049adantrl 716 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐶𝐵)) → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))) = (((𝐴o 𝐶) ·o 1o) +o ((𝐴 CNF 𝐵)‘(𝐵 × {∅}))))
51 oecl 8575 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
523, 51sylan 580 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
53 om1 8580 . . . . . . . . . . . . . . 15 ((𝐴o 𝐶) ∈ On → ((𝐴o 𝐶) ·o 1o) = (𝐴o 𝐶))
5452, 53syl 17 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ((𝐴o 𝐶) ·o 1o) = (𝐴o 𝐶))
551, 3, 4, 15cantnf0 9715 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
5655adantr 480 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
5754, 56oveq12d 7449 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (((𝐴o 𝐶) ·o 1o) +o ((𝐴 CNF 𝐵)‘(𝐵 × {∅}))) = ((𝐴o 𝐶) +o ∅))
58 oa0 8554 . . . . . . . . . . . . . 14 ((𝐴o 𝐶) ∈ On → ((𝐴o 𝐶) +o ∅) = (𝐴o 𝐶))
5952, 58syl 17 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ((𝐴o 𝐶) +o ∅) = (𝐴o 𝐶))
6057, 59eqtrd 2777 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (((𝐴o 𝐶) ·o 1o) +o ((𝐴 CNF 𝐵)‘(𝐵 × {∅}))) = (𝐴o 𝐶))
6160adantrr 717 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐶𝐵)) → (((𝐴o 𝐶) ·o 1o) +o ((𝐴 CNF 𝐵)‘(𝐵 × {∅}))) = (𝐴o 𝐶))
6250, 61eqtrd 2777 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐶𝐵)) → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))) = (𝐴o 𝐶))
6362eleq2d 2827 . . . . . . . . 9 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐶𝐵)) → (((𝐴 CNF 𝐵)‘𝐹) ∈ ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))) ↔ ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶)))
6463exp32 420 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (𝐶𝐵 → (((𝐴 CNF 𝐵)‘𝐹) ∈ ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))) ↔ ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶)))))
6564adantrd 491 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → ((𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵)) → (𝐶𝐵 → (((𝐴 CNF 𝐵)‘𝐹) ∈ ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))) ↔ ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶)))))
6665imp31 417 . . . . . 6 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ 𝐶𝐵) → (((𝐴 CNF 𝐵)‘𝐹) ∈ ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))) ↔ ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶)))
6724, 27, 663bitrrd 306 . . . . 5 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ 𝐶𝐵) → (((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶) ↔ 𝐹{⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)))} (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))))
68 fveq1 6905 . . . . . . . . . . 11 (𝑎 = 𝐹 → (𝑎𝑐) = (𝐹𝑐))
6968eleq1d 2826 . . . . . . . . . 10 (𝑎 = 𝐹 → ((𝑎𝑐) ∈ (𝑏𝑐) ↔ (𝐹𝑐) ∈ (𝑏𝑐)))
70 fveq1 6905 . . . . . . . . . . . . 13 (𝑎 = 𝐹 → (𝑎𝑥) = (𝐹𝑥))
7170eqeq1d 2739 . . . . . . . . . . . 12 (𝑎 = 𝐹 → ((𝑎𝑥) = (𝑏𝑥) ↔ (𝐹𝑥) = (𝑏𝑥)))
7271imbi2d 340 . . . . . . . . . . 11 (𝑎 = 𝐹 → ((𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)) ↔ (𝑐𝑥 → (𝐹𝑥) = (𝑏𝑥))))
7372ralbidv 3178 . . . . . . . . . 10 (𝑎 = 𝐹 → (∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)) ↔ ∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = (𝑏𝑥))))
7469, 73anbi12d 632 . . . . . . . . 9 (𝑎 = 𝐹 → (((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥))) ↔ ((𝐹𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = (𝑏𝑥)))))
7574rexbidv 3179 . . . . . . . 8 (𝑎 = 𝐹 → (∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥))) ↔ ∃𝑐𝐵 ((𝐹𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = (𝑏𝑥)))))
76 fveq1 6905 . . . . . . . . . . 11 (𝑏 = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) → (𝑏𝑐) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐))
7776eleq2d 2827 . . . . . . . . . 10 (𝑏 = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) → ((𝐹𝑐) ∈ (𝑏𝑐) ↔ (𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐)))
78 fveq1 6905 . . . . . . . . . . . . 13 (𝑏 = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) → (𝑏𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥))
7978eqeq2d 2748 . . . . . . . . . . . 12 (𝑏 = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) → ((𝐹𝑥) = (𝑏𝑥) ↔ (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)))
8079imbi2d 340 . . . . . . . . . . 11 (𝑏 = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) → ((𝑐𝑥 → (𝐹𝑥) = (𝑏𝑥)) ↔ (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥))))
8180ralbidv 3178 . . . . . . . . . 10 (𝑏 = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) → (∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = (𝑏𝑥)) ↔ ∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥))))
8277, 81anbi12d 632 . . . . . . . . 9 (𝑏 = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) → (((𝐹𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = (𝑏𝑥))) ↔ ((𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)))))
8382rexbidv 3179 . . . . . . . 8 (𝑏 = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) → (∃𝑐𝐵 ((𝐹𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = (𝑏𝑥))) ↔ ∃𝑐𝐵 ((𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)))))
8475, 83, 5bropabg 43336 . . . . . . 7 (𝐹{⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)))} (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) ↔ ((𝐹 ∈ V ∧ (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) ∈ V) ∧ ∃𝑐𝐵 ((𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)))))
85 fveq2 6906 . . . . . . . . . . . . . . . . . 18 (𝑐 = 𝐶 → (𝐹𝑐) = (𝐹𝐶))
8685adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑐𝐵) → (𝐹𝑐) = (𝐹𝐶))
87 eqeq1 2741 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑐 → (𝑦 = 𝐶𝑐 = 𝐶))
8887ifbid 4549 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑐 → if(𝑦 = 𝐶, 1o, ∅) = if(𝑐 = 𝐶, 1o, ∅))
89 1oex 8516 . . . . . . . . . . . . . . . . . . . 20 1o ∈ V
9089, 43ifex 4576 . . . . . . . . . . . . . . . . . . 19 if(𝑐 = 𝐶, 1o, ∅) ∈ V
9188, 16, 90fvmpt 7016 . . . . . . . . . . . . . . . . . 18 (𝑐𝐵 → ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) = if(𝑐 = 𝐶, 1o, ∅))
92 iftrue 4531 . . . . . . . . . . . . . . . . . 18 (𝑐 = 𝐶 → if(𝑐 = 𝐶, 1o, ∅) = 1o)
9391, 92sylan9eqr 2799 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑐𝐵) → ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) = 1o)
9486, 93eleq12d 2835 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝐶𝑐𝐵) → ((𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) ↔ (𝐹𝐶) ∈ 1o))
95 el1o 8533 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝐶) ∈ 1o ↔ (𝐹𝐶) = ∅)
9695a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝐶𝑐𝐵) → ((𝐹𝐶) ∈ 1o ↔ (𝐹𝐶) = ∅))
9796biimpd 229 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑐𝐵) → ((𝐹𝐶) ∈ 1o → (𝐹𝐶) = ∅))
98 simpl 482 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑐𝐵) → 𝑐 = 𝐶)
9997, 98jctild 525 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝐶𝑐𝐵) → ((𝐹𝐶) ∈ 1o → (𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅)))
10094, 99sylbid 240 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑐𝐵) → ((𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) → (𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅)))
101100expimpd 453 . . . . . . . . . . . . . 14 (𝑐 = 𝐶 → ((𝑐𝐵 ∧ (𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐)) → (𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅)))
10291adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑐𝐶𝑐𝐵) → ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) = if(𝑐 = 𝐶, 1o, ∅))
103 simpl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐𝐶𝑐𝐵) → 𝑐𝐶)
104103neneqd 2945 . . . . . . . . . . . . . . . . . . . 20 ((𝑐𝐶𝑐𝐵) → ¬ 𝑐 = 𝐶)
105104iffalsed 4536 . . . . . . . . . . . . . . . . . . 19 ((𝑐𝐶𝑐𝐵) → if(𝑐 = 𝐶, 1o, ∅) = ∅)
106102, 105eqtrd 2777 . . . . . . . . . . . . . . . . . 18 ((𝑐𝐶𝑐𝐵) → ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) = ∅)
107106eleq2d 2827 . . . . . . . . . . . . . . . . 17 ((𝑐𝐶𝑐𝐵) → ((𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) ↔ (𝐹𝑐) ∈ ∅))
108107biimpd 229 . . . . . . . . . . . . . . . 16 ((𝑐𝐶𝑐𝐵) → ((𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) → (𝐹𝑐) ∈ ∅))
109108expimpd 453 . . . . . . . . . . . . . . 15 (𝑐𝐶 → ((𝑐𝐵 ∧ (𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐)) → (𝐹𝑐) ∈ ∅))
110 noel 4338 . . . . . . . . . . . . . . . 16 ¬ (𝐹𝑐) ∈ ∅
111110pm2.21i 119 . . . . . . . . . . . . . . 15 ((𝐹𝑐) ∈ ∅ → (𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅))
112109, 111syl6 35 . . . . . . . . . . . . . 14 (𝑐𝐶 → ((𝑐𝐵 ∧ (𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐)) → (𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅)))
113101, 112pm2.61ine 3025 . . . . . . . . . . . . 13 ((𝑐𝐵 ∧ (𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐)) → (𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅))
114113a1i 11 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) → ((𝑐𝐵 ∧ (𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐)) → (𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅)))
115 fveqeq2 6915 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐶 → ((𝐹𝑥) = ∅ ↔ (𝐹𝐶) = ∅))
116115ralsng 4675 . . . . . . . . . . . . . . 15 (𝐶𝐵 → (∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅ ↔ (𝐹𝐶) = ∅))
117116anbi2d 630 . . . . . . . . . . . . . 14 (𝐶𝐵 → ((𝑐 = 𝐶 ∧ ∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅) ↔ (𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅)))
118117biimprd 248 . . . . . . . . . . . . 13 (𝐶𝐵 → ((𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅) → (𝑐 = 𝐶 ∧ ∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅)))
119118adantl 481 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) → ((𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅) → (𝑐 = 𝐶 ∧ ∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅)))
1204anim1i 615 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐵 ∈ On ∧ 𝐶 ∈ On))
121120adantr 480 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) → (𝐵 ∈ On ∧ 𝐶 ∈ On))
122 pm3.31 449 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝐵 → (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥))) → ((𝑥𝐵𝑐𝑥) → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)))
123122a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → ((𝑥𝐵 → (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥))) → ((𝑥𝐵𝑐𝑥) → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥))))
124 eldif 3961 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝐵 ∖ suc 𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥 ∈ suc 𝐶))
125 simplr 769 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥𝐵) → 𝑐 = 𝐶)
126125eleq1d 2826 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥𝐵) → (𝑐𝑥𝐶𝑥))
127 simpl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ∈ On)
128127adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → 𝐵 ∈ On)
129 onelon 6409 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
130128, 129sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥𝐵) → 𝑥 ∈ On)
131 simpllr 776 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥𝐵) → 𝐶 ∈ On)
132 ontri1 6418 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ On ∧ 𝐶 ∈ On) → (𝑥𝐶 ↔ ¬ 𝐶𝑥))
133130, 131, 132syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥𝐵) → (𝑥𝐶 ↔ ¬ 𝐶𝑥))
134133con2bid 354 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥𝐵) → (𝐶𝑥 ↔ ¬ 𝑥𝐶))
135 onsssuc 6474 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ On ∧ 𝐶 ∈ On) → (𝑥𝐶𝑥 ∈ suc 𝐶))
136130, 131, 135syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥𝐵) → (𝑥𝐶𝑥 ∈ suc 𝐶))
137136notbid 318 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥𝐵) → (¬ 𝑥𝐶 ↔ ¬ 𝑥 ∈ suc 𝐶))
138126, 134, 1373bitrrd 306 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥𝐵) → (¬ 𝑥 ∈ suc 𝐶𝑐𝑥))
139138pm5.32da 579 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → ((𝑥𝐵 ∧ ¬ 𝑥 ∈ suc 𝐶) ↔ (𝑥𝐵𝑐𝑥)))
140124, 139bitrid 283 . . . . . . . . . . . . . . . . . . . 20 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → (𝑥 ∈ (𝐵 ∖ suc 𝐶) ↔ (𝑥𝐵𝑐𝑥)))
141140biimpd 229 . . . . . . . . . . . . . . . . . . 19 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → (𝑥 ∈ (𝐵 ∖ suc 𝐶) → (𝑥𝐵𝑐𝑥)))
142141imim1d 82 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → (((𝑥𝐵𝑐𝑥) → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)) → (𝑥 ∈ (𝐵 ∖ suc 𝐶) → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥))))
143 eldifi 4131 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ (𝐵 ∖ suc 𝐶) → 𝑥𝐵)
144143adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → 𝑥𝐵)
145 eqeq1 2741 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑥 → (𝑦 = 𝐶𝑥 = 𝐶))
146145ifbid 4549 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑥 → if(𝑦 = 𝐶, 1o, ∅) = if(𝑥 = 𝐶, 1o, ∅))
14789, 43ifex 4576 . . . . . . . . . . . . . . . . . . . . . . . . 25 if(𝑥 = 𝐶, 1o, ∅) ∈ V
148146, 16, 147fvmpt 7016 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥𝐵 → ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥) = if(𝑥 = 𝐶, 1o, ∅))
149144, 148syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥) = if(𝑥 = 𝐶, 1o, ∅))
150128, 143, 129syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → 𝑥 ∈ On)
151 eloni 6394 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ On → Ord 𝑥)
152150, 151syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → Ord 𝑥)
153 eloni 6394 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐵 ∈ On → Ord 𝐵)
154153ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → Ord 𝐵)
155 simplr 769 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → 𝐶 ∈ On)
156 ordeldifsucon 43272 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Ord 𝐵𝐶 ∈ On) → (𝑥 ∈ (𝐵 ∖ suc 𝐶) ↔ (𝑥𝐵𝐶𝑥)))
157154, 155, 156syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → (𝑥 ∈ (𝐵 ∖ suc 𝐶) ↔ (𝑥𝐵𝐶𝑥)))
158157biimpa 476 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → (𝑥𝐵𝐶𝑥))
159 ordirr 6402 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Ord 𝑥 → ¬ 𝑥𝑥)
160 eleq1 2829 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = 𝐶 → (𝑥𝑥𝐶𝑥))
161160notbid 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 𝐶 → (¬ 𝑥𝑥 ↔ ¬ 𝐶𝑥))
162159, 161syl5ibcom 245 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Ord 𝑥 → (𝑥 = 𝐶 → ¬ 𝐶𝑥))
163162con2d 134 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Ord 𝑥 → (𝐶𝑥 → ¬ 𝑥 = 𝐶))
164163adantld 490 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Ord 𝑥 → ((𝑥𝐵𝐶𝑥) → ¬ 𝑥 = 𝐶))
165152, 158, 164sylc 65 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → ¬ 𝑥 = 𝐶)
166165iffalsed 4536 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → if(𝑥 = 𝐶, 1o, ∅) = ∅)
167149, 166eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥) = ∅)
168167eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → ((𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥) ↔ (𝐹𝑥) = ∅))
169168biimpd 229 . . . . . . . . . . . . . . . . . . . 20 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → ((𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥) → (𝐹𝑥) = ∅))
170169ex 412 . . . . . . . . . . . . . . . . . . 19 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → (𝑥 ∈ (𝐵 ∖ suc 𝐶) → ((𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥) → (𝐹𝑥) = ∅)))
171170a2d 29 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → ((𝑥 ∈ (𝐵 ∖ suc 𝐶) → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)) → (𝑥 ∈ (𝐵 ∖ suc 𝐶) → (𝐹𝑥) = ∅)))
172123, 142, 1713syld 60 . . . . . . . . . . . . . . . . 17 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → ((𝑥𝐵 → (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥))) → (𝑥 ∈ (𝐵 ∖ suc 𝐶) → (𝐹𝑥) = ∅)))
173172ralimdv2 3163 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → (∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)) → ∀𝑥 ∈ (𝐵 ∖ suc 𝐶)(𝐹𝑥) = ∅))
174121, 173sylan 580 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) ∧ 𝑐 = 𝐶) → (∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)) → ∀𝑥 ∈ (𝐵 ∖ suc 𝐶)(𝐹𝑥) = ∅))
175174adantr 480 . . . . . . . . . . . . . 14 ((((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) ∧ 𝑐 = 𝐶) ∧ ∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅) → (∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)) → ∀𝑥 ∈ (𝐵 ∖ suc 𝐶)(𝐹𝑥) = ∅))
176 ralun 4198 . . . . . . . . . . . . . . . . 17 ((∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅ ∧ ∀𝑥 ∈ (𝐵 ∖ suc 𝐶)(𝐹𝑥) = ∅) → ∀𝑥 ∈ ({𝐶} ∪ (𝐵 ∖ suc 𝐶))(𝐹𝑥) = ∅)
177176adantll 714 . . . . . . . . . . . . . . . 16 (((((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) ∧ 𝑐 = 𝐶) ∧ ∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅) ∧ ∀𝑥 ∈ (𝐵 ∖ suc 𝐶)(𝐹𝑥) = ∅) → ∀𝑥 ∈ ({𝐶} ∪ (𝐵 ∖ suc 𝐶))(𝐹𝑥) = ∅)
178 undif3 4300 . . . . . . . . . . . . . . . . . . . . 21 ({𝐶} ∪ (𝐵 ∖ suc 𝐶)) = (({𝐶} ∪ 𝐵) ∖ (suc 𝐶 ∖ {𝐶}))
179 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐶 ∈ On ∧ 𝐶𝐵) → 𝐶𝐵)
180179snssd 4809 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ∈ On ∧ 𝐶𝐵) → {𝐶} ⊆ 𝐵)
181 ssequn1 4186 . . . . . . . . . . . . . . . . . . . . . . 23 ({𝐶} ⊆ 𝐵 ↔ ({𝐶} ∪ 𝐵) = 𝐵)
182180, 181sylib 218 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 ∈ On ∧ 𝐶𝐵) → ({𝐶} ∪ 𝐵) = 𝐵)
183 simpl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐶 ∈ On ∧ 𝐶𝐵) → 𝐶 ∈ On)
184 eloni 6394 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐶 ∈ On → Ord 𝐶)
185 orddif 6480 . . . . . . . . . . . . . . . . . . . . . . . 24 (Ord 𝐶𝐶 = (suc 𝐶 ∖ {𝐶}))
186183, 184, 1853syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ∈ On ∧ 𝐶𝐵) → 𝐶 = (suc 𝐶 ∖ {𝐶}))
187186eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 ∈ On ∧ 𝐶𝐵) → (suc 𝐶 ∖ {𝐶}) = 𝐶)
188182, 187difeq12d 4127 . . . . . . . . . . . . . . . . . . . . 21 ((𝐶 ∈ On ∧ 𝐶𝐵) → (({𝐶} ∪ 𝐵) ∖ (suc 𝐶 ∖ {𝐶})) = (𝐵𝐶))
189178, 188eqtrid 2789 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 ∈ On ∧ 𝐶𝐵) → ({𝐶} ∪ (𝐵 ∖ suc 𝐶)) = (𝐵𝐶))
190189adantll 714 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) → ({𝐶} ∪ (𝐵 ∖ suc 𝐶)) = (𝐵𝐶))
191190adantr 480 . . . . . . . . . . . . . . . . . 18 (((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) ∧ 𝑐 = 𝐶) → ({𝐶} ∪ (𝐵 ∖ suc 𝐶)) = (𝐵𝐶))
192191raleqdv 3326 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) ∧ 𝑐 = 𝐶) → (∀𝑥 ∈ ({𝐶} ∪ (𝐵 ∖ suc 𝐶))(𝐹𝑥) = ∅ ↔ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅))
193192ad2antrr 726 . . . . . . . . . . . . . . . 16 (((((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) ∧ 𝑐 = 𝐶) ∧ ∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅) ∧ ∀𝑥 ∈ (𝐵 ∖ suc 𝐶)(𝐹𝑥) = ∅) → (∀𝑥 ∈ ({𝐶} ∪ (𝐵 ∖ suc 𝐶))(𝐹𝑥) = ∅ ↔ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅))
194177, 193mpbid 232 . . . . . . . . . . . . . . 15 (((((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) ∧ 𝑐 = 𝐶) ∧ ∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅) ∧ ∀𝑥 ∈ (𝐵 ∖ suc 𝐶)(𝐹𝑥) = ∅) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅)
195194ex 412 . . . . . . . . . . . . . 14 ((((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) ∧ 𝑐 = 𝐶) ∧ ∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅) → (∀𝑥 ∈ (𝐵 ∖ suc 𝐶)(𝐹𝑥) = ∅ → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅))
196175, 195syld 47 . . . . . . . . . . . . 13 ((((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) ∧ 𝑐 = 𝐶) ∧ ∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅) → (∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅))
197196expl 457 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) → ((𝑐 = 𝐶 ∧ ∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅) → (∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅)))
198114, 119, 1973syld 60 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) → ((𝑐𝐵 ∧ (𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐)) → (∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅)))
199198expdimp 452 . . . . . . . . . 10 (((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) ∧ 𝑐𝐵) → ((𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) → (∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅)))
200199impd 410 . . . . . . . . 9 (((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) ∧ 𝑐𝐵) → (((𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥))) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅))
201200rexlimdva 3155 . . . . . . . 8 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) → (∃𝑐𝐵 ((𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥))) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅))
202201adantld 490 . . . . . . 7 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) → (((𝐹 ∈ V ∧ (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) ∈ V) ∧ ∃𝑐𝐵 ((𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)))) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅))
20384, 202biimtrid 242 . . . . . 6 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) → (𝐹{⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)))} (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅))
204203adantlrr 721 . . . . 5 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ 𝐶𝐵) → (𝐹{⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)))} (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅))
20567, 204sylbid 240 . . . 4 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ 𝐶𝐵) → (((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅))
206205ex 412 . . 3 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (𝐶𝐵 → (((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅)))
207 ral0 4513 . . . . 5 𝑥 ∈ ∅ (𝐹𝑥) = ∅
208 ssdif0 4366 . . . . . . 7 (𝐵𝐶 ↔ (𝐵𝐶) = ∅)
209208biimpi 216 . . . . . 6 (𝐵𝐶 → (𝐵𝐶) = ∅)
210209raleqdv 3326 . . . . 5 (𝐵𝐶 → (∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅ ↔ ∀𝑥 ∈ ∅ (𝐹𝑥) = ∅))
211207, 210mpbiri 258 . . . 4 (𝐵𝐶 → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅)
212211a1i13 27 . . 3 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (𝐵𝐶 → (((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅)))
213184adantr 480 . . . 4 ((𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵)) → Ord 𝐶)
214153adantl 481 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → Ord 𝐵)
215 ordtri2or 6482 . . . 4 ((Ord 𝐶 ∧ Ord 𝐵) → (𝐶𝐵𝐵𝐶))
216213, 214, 215syl2anr 597 . . 3 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (𝐶𝐵𝐵𝐶))
217206, 212, 216mpjaod 861 . 2 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅))
2183ad2antrr 726 . . . 4 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → 𝐴 ∈ On)
219 simpllr 776 . . . 4 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → 𝐵 ∈ On)
220 simplrr 778 . . . 4 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → 𝐹 ∈ dom (𝐴 CNF 𝐵))
22115ad2antrr 726 . . . 4 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → ∅ ∈ 𝐴)
222 simplrl 777 . . . 4 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → 𝐶 ∈ On)
2231, 3, 4cantnfs 9706 . . . . . . . . . 10 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝐹 ∈ dom (𝐴 CNF 𝐵) ↔ (𝐹:𝐵𝐴𝐹 finSupp ∅)))
224223biimpd 229 . . . . . . . . 9 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝐹 ∈ dom (𝐴 CNF 𝐵) → (𝐹:𝐵𝐴𝐹 finSupp ∅)))
225224adantld 490 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → ((𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵)) → (𝐹:𝐵𝐴𝐹 finSupp ∅)))
226225imp 406 . . . . . . 7 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (𝐹:𝐵𝐴𝐹 finSupp ∅))
227226simpld 494 . . . . . 6 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → 𝐹:𝐵𝐴)
228227adantr 480 . . . . 5 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → 𝐹:𝐵𝐴)
229 fveqeq2 6915 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐹𝑥) = ∅ ↔ (𝐹𝑦) = ∅))
230229rspccv 3619 . . . . . . 7 (∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅ → (𝑦 ∈ (𝐵𝐶) → (𝐹𝑦) = ∅))
231230adantl 481 . . . . . 6 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → (𝑦 ∈ (𝐵𝐶) → (𝐹𝑦) = ∅))
232231imp 406 . . . . 5 (((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) ∧ 𝑦 ∈ (𝐵𝐶)) → (𝐹𝑦) = ∅)
233228, 232suppss 8219 . . . 4 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → (𝐹 supp ∅) ⊆ 𝐶)
2341, 218, 219, 220, 221, 222, 233cantnflt2 9713 . . 3 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶))
235234ex 412 . 2 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅ → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶)))
236217, 235impbid 212 1 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶) ↔ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  cun 3949  wss 3951  c0 4333  ifcif 4525  {csn 4626   class class class wbr 5143  {copab 5205  cmpt 5225   E cep 5583   × cxp 5683  dom cdm 5685  Ord word 6383  Oncon0 6384  suc csuc 6386  wf 6557  cfv 6561   Isom wiso 6562  (class class class)co 7431   supp csupp 8185  1oc1o 8499  2oc2o 8500   +o coa 8503   ·o comu 8504  o coe 8505   finSupp cfsupp 9401   CNF ccnf 9701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqom 8488  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-oexp 8512  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-cnf 9702
This theorem is referenced by:  cantnf2  43338
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