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Theorem cantnfresb 42007
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 2733 . . . . . . . . . . 11 dom (𝐴 CNF 𝐵) = dom (𝐴 CNF 𝐵)
2 eldifi 4125 . . . . . . . . . . . 12 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
32adantr 482 . . . . . . . . . . 11 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
4 simpr 486 . . . . . . . . . . 11 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
5 eqid 2733 . . . . . . . . . . 11 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)))} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝐵 ((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑥𝐵 (𝑐𝑥 → (𝑎𝑥) = (𝑏𝑥)))}
61, 3, 4, 5cantnf 9684 . . . . . . . . . 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 482 . . . . . . . . 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 486 . . . . . . . . 9 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵)) → 𝐹 ∈ dom (𝐴 CNF 𝐵))
9 ondif2 8497 . . . . . . . . . . . . . . 15 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
109simprbi 498 . . . . . . . . . . . . . 14 (𝐴 ∈ (On ∖ 2o) → 1o𝐴)
11 dif20el 8500 . . . . . . . . . . . . . 14 (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
1210, 11ifcld 4573 . . . . . . . . . . . . 13 (𝐴 ∈ (On ∖ 2o) → if(𝑦 = 𝐶, 1o, ∅) ∈ 𝐴)
1312ad2antrr 725 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝑦𝐵) → if(𝑦 = 𝐶, 1o, ∅) ∈ 𝐴)
1413fmpttd 7110 . . . . . . . . . . 11 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)):𝐵𝐴)
1511adantr 482 . . . . . . . . . . . 12 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → ∅ ∈ 𝐴)
16 eqid 2733 . . . . . . . . . . . 12 (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))
174, 15, 16sniffsupp 9391 . . . . . . . . . . 11 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) finSupp ∅)
181, 3, 4cantnfs 9657 . . . . . . . . . . 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 712 . . . . . . . . . 10 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) ∈ dom (𝐴 CNF 𝐵))
2019adantr 482 . . . . . . . . 9 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵)) → (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) ∈ dom (𝐴 CNF 𝐵))
21 isorel 7318 . . . . . . . . 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 836 . . . . . . . 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 715 . . . . . . 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 482 . . . . . 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 6903 . . . . . . 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 5580 . . . . . . 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 725 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶𝐵) → 𝐴 ∈ On)
29 simplr 768 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶𝐵) → 𝐵 ∈ On)
30 fconst6g 6777 . . . . . . . . . . . . . . . . . 18 (∅ ∈ 𝐴 → (𝐵 × {∅}):𝐵𝐴)
3111, 30syl 17 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ (On ∖ 2o) → (𝐵 × {∅}):𝐵𝐴)
3231adantr 482 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝐵 × {∅}):𝐵𝐴)
334, 15fczfsuppd 9377 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝐵 × {∅}) finSupp ∅)
341, 3, 4cantnfs 9657 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → ((𝐵 × {∅}) ∈ dom (𝐴 CNF 𝐵) ↔ ((𝐵 × {∅}):𝐵𝐴 ∧ (𝐵 × {∅}) finSupp ∅)))
3532, 33, 34mpbir2and 712 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝐵 × {∅}) ∈ dom (𝐴 CNF 𝐵))
3635adantr 482 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶𝐵) → (𝐵 × {∅}) ∈ dom (𝐴 CNF 𝐵))
37 simpr 486 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶𝐵) → 𝐶𝐵)
3810ad2antrr 725 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶𝐵) → 1o𝐴)
39 fczsupp0 8173 . . . . . . . . . . . . . . . 16 ((𝐵 × {∅}) supp ∅) = ∅
40 0ss 4395 . . . . . . . . . . . . . . . 16 ∅ ⊆ 𝐶
4139, 40eqsstri 4015 . . . . . . . . . . . . . . 15 ((𝐵 × {∅}) supp ∅) ⊆ 𝐶
4241a1i 11 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶𝐵) → ((𝐵 × {∅}) supp ∅) ⊆ 𝐶)
43 0ex 5306 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
4443fvconst2 7200 . . . . . . . . . . . . . . . . 17 (𝑦𝐵 → ((𝐵 × {∅})‘𝑦) = ∅)
4544ifeq2d 4547 . . . . . . . . . . . . . . . 16 (𝑦𝐵 → if(𝑦 = 𝐶, 1o, ((𝐵 × {∅})‘𝑦)) = if(𝑦 = 𝐶, 1o, ∅))
4645mpteq2ia 5250 . . . . . . . . . . . . . . 15 (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ((𝐵 × {∅})‘𝑦))) = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))
4746eqcomi 2742 . . . . . . . . . . . . . 14 (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ((𝐵 × {∅})‘𝑦)))
481, 28, 29, 36, 37, 38, 42, 47cantnfp1 9672 . . . . . . . . . . . . 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 497 . . . . . . . . . . . 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 715 . . . . . . . . . . 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 8532 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
523, 51sylan 581 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
53 om1 8538 . . . . . . . . . . . . . . 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 9666 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
5655adantr 482 . . . . . . . . . . . . . 14 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
5754, 56oveq12d 7422 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (((𝐴o 𝐶) ·o 1o) +o ((𝐴 CNF 𝐵)‘(𝐵 × {∅}))) = ((𝐴o 𝐶) +o ∅))
58 oa0 8511 . . . . . . . . . . . . . 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 2773 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (((𝐴o 𝐶) ·o 1o) +o ((𝐴 CNF 𝐵)‘(𝐵 × {∅}))) = (𝐴o 𝐶))
6160adantrr 716 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐶𝐵)) → (((𝐴o 𝐶) ·o 1o) +o ((𝐴 CNF 𝐵)‘(𝐵 × {∅}))) = (𝐴o 𝐶))
6250, 61eqtrd 2773 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐶𝐵)) → ((𝐴 CNF 𝐵)‘(𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))) = (𝐴o 𝐶))
6362eleq2d 2820 . . . . . . . . 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 422 . . . . . . . 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 493 . . . . . . 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 419 . . . . . 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 6887 . . . . . . . . . . 11 (𝑎 = 𝐹 → (𝑎𝑐) = (𝐹𝑐))
6968eleq1d 2819 . . . . . . . . . 10 (𝑎 = 𝐹 → ((𝑎𝑐) ∈ (𝑏𝑐) ↔ (𝐹𝑐) ∈ (𝑏𝑐)))
70 fveq1 6887 . . . . . . . . . . . . 13 (𝑎 = 𝐹 → (𝑎𝑥) = (𝐹𝑥))
7170eqeq1d 2735 . . . . . . . . . . . 12 (𝑎 = 𝐹 → ((𝑎𝑥) = (𝑏𝑥) ↔ (𝐹𝑥) = (𝑏𝑥)))
7271imbi2d 341 . . . . . . . . . . 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 6887 . . . . . . . . . . 11 (𝑏 = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) → (𝑏𝑐) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐))
7776eleq2d 2820 . . . . . . . . . 10 (𝑏 = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) → ((𝐹𝑐) ∈ (𝑏𝑐) ↔ (𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐)))
78 fveq1 6887 . . . . . . . . . . . . 13 (𝑏 = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) → (𝑏𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥))
7978eqeq2d 2744 . . . . . . . . . . . 12 (𝑏 = (𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅)) → ((𝐹𝑥) = (𝑏𝑥) ↔ (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)))
8079imbi2d 341 . . . . . . . . . . 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 42006 . . . . . . 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 6888 . . . . . . . . . . . . . . . . . 18 (𝑐 = 𝐶 → (𝐹𝑐) = (𝐹𝐶))
8685adantr 482 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑐𝐵) → (𝐹𝑐) = (𝐹𝐶))
87 eqeq1 2737 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑐 → (𝑦 = 𝐶𝑐 = 𝐶))
8887ifbid 4550 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑐 → if(𝑦 = 𝐶, 1o, ∅) = if(𝑐 = 𝐶, 1o, ∅))
89 1oex 8471 . . . . . . . . . . . . . . . . . . . 20 1o ∈ V
9089, 43ifex 4577 . . . . . . . . . . . . . . . . . . 19 if(𝑐 = 𝐶, 1o, ∅) ∈ V
9188, 16, 90fvmpt 6994 . . . . . . . . . . . . . . . . . 18 (𝑐𝐵 → ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) = if(𝑐 = 𝐶, 1o, ∅))
92 iftrue 4533 . . . . . . . . . . . . . . . . . 18 (𝑐 = 𝐶 → if(𝑐 = 𝐶, 1o, ∅) = 1o)
9391, 92sylan9eqr 2795 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑐𝐵) → ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) = 1o)
9486, 93eleq12d 2828 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝐶𝑐𝐵) → ((𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) ↔ (𝐹𝐶) ∈ 1o))
95 el1o 8490 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝐶) ∈ 1o ↔ (𝐹𝐶) = ∅)
9695a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝐶𝑐𝐵) → ((𝐹𝐶) ∈ 1o ↔ (𝐹𝐶) = ∅))
9796biimpd 228 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑐𝐵) → ((𝐹𝐶) ∈ 1o → (𝐹𝐶) = ∅))
98 simpl 484 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑐𝐵) → 𝑐 = 𝐶)
9997, 98jctild 527 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝐶𝑐𝐵) → ((𝐹𝐶) ∈ 1o → (𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅)))
10094, 99sylbid 239 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑐𝐵) → ((𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) → (𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅)))
101100expimpd 455 . . . . . . . . . . . . . 14 (𝑐 = 𝐶 → ((𝑐𝐵 ∧ (𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐)) → (𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅)))
10291adantl 483 . . . . . . . . . . . . . . . . . . 19 ((𝑐𝐶𝑐𝐵) → ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) = if(𝑐 = 𝐶, 1o, ∅))
103 simpl 484 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐𝐶𝑐𝐵) → 𝑐𝐶)
104103neneqd 2946 . . . . . . . . . . . . . . . . . . . 20 ((𝑐𝐶𝑐𝐵) → ¬ 𝑐 = 𝐶)
105104iffalsed 4538 . . . . . . . . . . . . . . . . . . 19 ((𝑐𝐶𝑐𝐵) → if(𝑐 = 𝐶, 1o, ∅) = ∅)
106102, 105eqtrd 2773 . . . . . . . . . . . . . . . . . 18 ((𝑐𝐶𝑐𝐵) → ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) = ∅)
107106eleq2d 2820 . . . . . . . . . . . . . . . . 17 ((𝑐𝐶𝑐𝐵) → ((𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) ↔ (𝐹𝑐) ∈ ∅))
108107biimpd 228 . . . . . . . . . . . . . . . 16 ((𝑐𝐶𝑐𝐵) → ((𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐) → (𝐹𝑐) ∈ ∅))
109108expimpd 455 . . . . . . . . . . . . . . 15 (𝑐𝐶 → ((𝑐𝐵 ∧ (𝐹𝑐) ∈ ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑐)) → (𝐹𝑐) ∈ ∅))
110 noel 4329 . . . . . . . . . . . . . . . 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 3026 . . . . . . . . . . . . 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 6897 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐶 → ((𝐹𝑥) = ∅ ↔ (𝐹𝐶) = ∅))
116115ralsng 4676 . . . . . . . . . . . . . . 15 (𝐶𝐵 → (∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅ ↔ (𝐹𝐶) = ∅))
117116anbi2d 630 . . . . . . . . . . . . . 14 (𝐶𝐵 → ((𝑐 = 𝐶 ∧ ∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅) ↔ (𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅)))
118117biimprd 247 . . . . . . . . . . . . 13 (𝐶𝐵 → ((𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅) → (𝑐 = 𝐶 ∧ ∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅)))
119118adantl 483 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) → ((𝑐 = 𝐶 ∧ (𝐹𝐶) = ∅) → (𝑐 = 𝐶 ∧ ∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅)))
1204anim1i 616 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐵 ∈ On ∧ 𝐶 ∈ On))
121120adantr 482 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) → (𝐵 ∈ On ∧ 𝐶 ∈ On))
122 pm3.31 451 . . . . . . . . . . . . . . . . . . 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 3957 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝐵 ∖ suc 𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥 ∈ suc 𝐶))
125 simplr 768 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥𝐵) → 𝑐 = 𝐶)
126125eleq1d 2819 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥𝐵) → (𝑐𝑥𝐶𝑥))
127 simpl 484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ∈ On)
128127adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → 𝐵 ∈ On)
129 onelon 6386 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
130128, 129sylan 581 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥𝐵) → 𝑥 ∈ On)
131 simpllr 775 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥𝐵) → 𝐶 ∈ On)
132 ontri1 6395 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ On ∧ 𝐶 ∈ On) → (𝑥𝐶 ↔ ¬ 𝐶𝑥))
133130, 131, 132syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥𝐵) → (𝑥𝐶 ↔ ¬ 𝐶𝑥))
134133con2bid 355 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥𝐵) → (𝐶𝑥 ↔ ¬ 𝑥𝐶))
135 onsssuc 6451 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ On ∧ 𝐶 ∈ On) → (𝑥𝐶𝑥 ∈ suc 𝐶))
136130, 131, 135syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . . . . . . . . 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 228 . . . . . . . . . . . . . . . . . . 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 4125 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ (𝐵 ∖ suc 𝐶) → 𝑥𝐵)
144143adantl 483 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → 𝑥𝐵)
145 eqeq1 2737 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑥 → (𝑦 = 𝐶𝑥 = 𝐶))
146145ifbid 4550 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑥 → if(𝑦 = 𝐶, 1o, ∅) = if(𝑥 = 𝐶, 1o, ∅))
14789, 43ifex 4577 . . . . . . . . . . . . . . . . . . . . . . . . 25 if(𝑥 = 𝐶, 1o, ∅) ∈ V
148146, 16, 147fvmpt 6994 . . . . . . . . . . . . . . . . . . . . . . . 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 597 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → 𝑥 ∈ On)
151 eloni 6371 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ On → Ord 𝑥)
152150, 151syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → Ord 𝑥)
153 eloni 6371 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐵 ∈ On → Ord 𝐵)
154153ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → Ord 𝐵)
155 simplr 768 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → 𝐶 ∈ On)
156 ordeldifsucon 41942 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Ord 𝐵𝐶 ∈ On) → (𝑥 ∈ (𝐵 ∖ suc 𝐶) ↔ (𝑥𝐵𝐶𝑥)))
157154, 155, 156syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → (𝑥 ∈ (𝐵 ∖ suc 𝐶) ↔ (𝑥𝐵𝐶𝑥)))
158157biimpa 478 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → (𝑥𝐵𝐶𝑥))
159 ordirr 6379 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Ord 𝑥 → ¬ 𝑥𝑥)
160 eleq1 2822 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = 𝐶 → (𝑥𝑥𝐶𝑥))
161160notbid 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 𝐶 → (¬ 𝑥𝑥 ↔ ¬ 𝐶𝑥))
162159, 161syl5ibcom 244 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Ord 𝑥 → (𝑥 = 𝐶 → ¬ 𝐶𝑥))
163162con2d 134 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Ord 𝑥 → (𝐶𝑥 → ¬ 𝑥 = 𝐶))
164163adantld 492 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Ord 𝑥 → ((𝑥𝐵𝐶𝑥) → ¬ 𝑥 = 𝐶))
165152, 158, 164sylc 65 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → ¬ 𝑥 = 𝐶)
166165iffalsed 4538 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → if(𝑥 = 𝐶, 1o, ∅) = ∅)
167149, 166eqtrd 2773 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥) = ∅)
168167eqeq2d 2744 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → ((𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥) ↔ (𝐹𝑥) = ∅))
169168biimpd 228 . . . . . . . . . . . . . . . . . . . 20 ((((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) ∧ 𝑥 ∈ (𝐵 ∖ suc 𝐶)) → ((𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥) → (𝐹𝑥) = ∅))
170169ex 414 . . . . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑐 = 𝐶) → (∀𝑥𝐵 (𝑐𝑥 → (𝐹𝑥) = ((𝑦𝐵 ↦ if(𝑦 = 𝐶, 1o, ∅))‘𝑥)) → ∀𝑥 ∈ (𝐵 ∖ suc 𝐶)(𝐹𝑥) = ∅))
174121, 173sylan 581 . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . 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 4191 . . . . . . . . . . . . . . . . 17 ((∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅ ∧ ∀𝑥 ∈ (𝐵 ∖ suc 𝐶)(𝐹𝑥) = ∅) → ∀𝑥 ∈ ({𝐶} ∪ (𝐵 ∖ suc 𝐶))(𝐹𝑥) = ∅)
177176adantll 713 . . . . . . . . . . . . . . . 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 4289 . . . . . . . . . . . . . . . . . . . . 21 ({𝐶} ∪ (𝐵 ∖ suc 𝐶)) = (({𝐶} ∪ 𝐵) ∖ (suc 𝐶 ∖ {𝐶}))
179 simpr 486 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐶 ∈ On ∧ 𝐶𝐵) → 𝐶𝐵)
180179snssd 4811 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ∈ On ∧ 𝐶𝐵) → {𝐶} ⊆ 𝐵)
181 ssequn1 4179 . . . . . . . . . . . . . . . . . . . . . . 23 ({𝐶} ⊆ 𝐵 ↔ ({𝐶} ∪ 𝐵) = 𝐵)
182180, 181sylib 217 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 ∈ On ∧ 𝐶𝐵) → ({𝐶} ∪ 𝐵) = 𝐵)
183 simpl 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐶 ∈ On ∧ 𝐶𝐵) → 𝐶 ∈ On)
184 eloni 6371 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐶 ∈ On → Ord 𝐶)
185 orddif 6457 . . . . . . . . . . . . . . . . . . . . . . . 24 (Ord 𝐶𝐶 = (suc 𝐶 ∖ {𝐶}))
186183, 184, 1853syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ∈ On ∧ 𝐶𝐵) → 𝐶 = (suc 𝐶 ∖ {𝐶}))
187186eqcomd 2739 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 ∈ On ∧ 𝐶𝐵) → (suc 𝐶 ∖ {𝐶}) = 𝐶)
188182, 187difeq12d 4122 . . . . . . . . . . . . . . . . . . . . 21 ((𝐶 ∈ On ∧ 𝐶𝐵) → (({𝐶} ∪ 𝐵) ∖ (suc 𝐶 ∖ {𝐶})) = (𝐵𝐶))
189178, 188eqtrid 2785 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 ∈ On ∧ 𝐶𝐵) → ({𝐶} ∪ (𝐵 ∖ suc 𝐶)) = (𝐵𝐶))
190189adantll 713 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) → ({𝐶} ∪ (𝐵 ∖ suc 𝐶)) = (𝐵𝐶))
191190adantr 482 . . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . 15 (((((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ∧ 𝐶𝐵) ∧ 𝑐 = 𝐶) ∧ ∀𝑥 ∈ {𝐶} (𝐹𝑥) = ∅) ∧ ∀𝑥 ∈ (𝐵 ∖ suc 𝐶)(𝐹𝑥) = ∅) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅)
195194ex 414 . . . . . . . . . . . . . 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 459 . . . . . . . . . . . 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 454 . . . . . . . . . 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 412 . . . . . . . . 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 3156 . . . . . . . 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 492 . . . . . . 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 241 . . . . . 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 720 . . . . 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 239 . . . 4 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ 𝐶𝐵) → (((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅))
206205ex 414 . . 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 4511 . . . . 5 𝑥 ∈ ∅ (𝐹𝑥) = ∅
208 ssdif0 4362 . . . . . . 7 (𝐵𝐶 ↔ (𝐵𝐶) = ∅)
209208biimpi 215 . . . . . 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 482 . . . 4 ((𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵)) → Ord 𝐶)
214153adantl 483 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → Ord 𝐵)
215 ordtri2or 6459 . . . 4 ((Ord 𝐶 ∧ Ord 𝐵) → (𝐶𝐵𝐵𝐶))
216213, 214, 215syl2anr 598 . . 3 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (𝐶𝐵𝐵𝐶))
217206, 212, 216mpjaod 859 . 2 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶) → ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅))
2183ad2antrr 725 . . . 4 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → 𝐴 ∈ On)
219 simpllr 775 . . . 4 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → 𝐵 ∈ On)
220 simplrr 777 . . . 4 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → 𝐹 ∈ dom (𝐴 CNF 𝐵))
22115ad2antrr 725 . . . 4 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → ∅ ∈ 𝐴)
222 simplrl 776 . . . 4 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → 𝐶 ∈ On)
2231, 3, 4cantnfs 9657 . . . . . . . . . 10 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝐹 ∈ dom (𝐴 CNF 𝐵) ↔ (𝐹:𝐵𝐴𝐹 finSupp ∅)))
224223biimpd 228 . . . . . . . . 9 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝐹 ∈ dom (𝐴 CNF 𝐵) → (𝐹:𝐵𝐴𝐹 finSupp ∅)))
225224adantld 492 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → ((𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵)) → (𝐹:𝐵𝐴𝐹 finSupp ∅)))
226225imp 408 . . . . . . 7 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (𝐹:𝐵𝐴𝐹 finSupp ∅))
227226simpld 496 . . . . . 6 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → 𝐹:𝐵𝐴)
228227adantr 482 . . . . 5 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → 𝐹:𝐵𝐴)
229 fveqeq2 6897 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐹𝑥) = ∅ ↔ (𝐹𝑦) = ∅))
230229rspccv 3609 . . . . . . 7 (∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅ → (𝑦 ∈ (𝐵𝐶) → (𝐹𝑦) = ∅))
231230adantl 483 . . . . . 6 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → (𝑦 ∈ (𝐵𝐶) → (𝐹𝑦) = ∅))
232231imp 408 . . . . 5 (((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) ∧ 𝑦 ∈ (𝐵𝐶)) → (𝐹𝑦) = ∅)
233228, 232suppss 8174 . . . 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 9664 . . 3 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) ∧ ∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅) → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶))
235234ex 414 . 2 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (∀𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = ∅ → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴o 𝐶)))
236217, 235impbid 211 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 205  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  Vcvv 3475  cdif 3944  cun 3945  wss 3947  c0 4321  ifcif 4527  {csn 4627   class class class wbr 5147  {copab 5209  cmpt 5230   E cep 5578   × cxp 5673  dom cdm 5675  Ord word 6360  Oncon0 6361  suc csuc 6363  wf 6536  cfv 6540   Isom wiso 6541  (class class class)co 7404   supp csupp 8141  1oc1o 8454  2oc2o 8455   +o coa 8458   ·o comu 8459  o coe 8460   finSupp cfsupp 9357   CNF ccnf 9652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-supp 8142  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-seqom 8443  df-1o 8461  df-2o 8462  df-oadd 8465  df-omul 8466  df-oexp 8467  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-oi 9501  df-cnf 9653
This theorem is referenced by:  cantnf2  42008
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