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Theorem cantnf2 43338
Description: For every ordinal, 𝐴, there is a an ordinal exponent 𝑏 such that 𝐴 is less than (ω ↑o 𝑏) and for every ordinal at least as large as 𝑏 there is a unique Cantor normal form, 𝑓, with zeros for all the unnecessary higher terms, that sums to 𝐴. Theorem 5.3 of [Schloeder] p. 16. (Contributed by RP, 3-Feb-2025.)
Assertion
Ref Expression
cantnf2 (𝐴 ∈ On → ∃𝑏 ∈ On ∀𝑐 ∈ (On ∖ 𝑏)∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))
Distinct variable group:   𝐴,𝑏,𝑐,𝑓

Proof of Theorem cantnf2
Dummy variables 𝑎 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onexoegt 43256 . 2 (𝐴 ∈ On → ∃𝑏 ∈ On 𝐴 ∈ (ω ↑o 𝑏))
2 eldif 3961 . . . . . . 7 (𝑐 ∈ (On ∖ 𝑏) ↔ (𝑐 ∈ On ∧ ¬ 𝑐𝑏))
3 simp2 1138 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) → 𝑏 ∈ On)
4 pm3.2 469 . . . . . . . . . 10 (𝑏 ∈ On → (𝑐 ∈ On → (𝑏 ∈ On ∧ 𝑐 ∈ On)))
53, 4syl 17 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) → (𝑐 ∈ On → (𝑏 ∈ On ∧ 𝑐 ∈ On)))
6 ontri1 6418 . . . . . . . . 9 ((𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑏𝑐 ↔ ¬ 𝑐𝑏))
75, 6syl6 35 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) → (𝑐 ∈ On → (𝑏𝑐 ↔ ¬ 𝑐𝑏)))
87pm5.32d 577 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) → ((𝑐 ∈ On ∧ 𝑏𝑐) ↔ (𝑐 ∈ On ∧ ¬ 𝑐𝑏)))
92, 8bitr4id 290 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) → (𝑐 ∈ (On ∖ 𝑏) ↔ (𝑐 ∈ On ∧ 𝑏𝑐)))
10 simplr 769 . . . . . . . . . . . 12 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → 𝑎 = 𝐴)
1110breq2d 5155 . . . . . . . . . . 11 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (𝑓(ω CNF 𝑐)𝑎𝑓(ω CNF 𝑐)𝐴))
12 eqid 2737 . . . . . . . . . . . . . 14 dom (ω CNF 𝑐) = dom (ω CNF 𝑐)
13 omelon 9686 . . . . . . . . . . . . . . 15 ω ∈ On
1413a1i 11 . . . . . . . . . . . . . 14 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → ω ∈ On)
15 simprl 771 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → 𝑐 ∈ On)
1615ad2antrr 726 . . . . . . . . . . . . . 14 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → 𝑐 ∈ On)
1712, 14, 16cantnff1o 9736 . . . . . . . . . . . . 13 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (ω CNF 𝑐):dom (ω CNF 𝑐)–1-1-onto→(ω ↑o 𝑐))
18 f1ofun 6850 . . . . . . . . . . . . 13 ((ω CNF 𝑐):dom (ω CNF 𝑐)–1-1-onto→(ω ↑o 𝑐) → Fun (ω CNF 𝑐))
1917, 18syl 17 . . . . . . . . . . . 12 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → Fun (ω CNF 𝑐))
20 funbrfvb 6962 . . . . . . . . . . . 12 ((Fun (ω CNF 𝑐) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (((ω CNF 𝑐)‘𝑓) = 𝐴𝑓(ω CNF 𝑐)𝐴))
2119, 20sylancom 588 . . . . . . . . . . 11 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (((ω CNF 𝑐)‘𝑓) = 𝐴𝑓(ω CNF 𝑐)𝐴))
2211, 21bitr4d 282 . . . . . . . . . 10 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (𝑓(ω CNF 𝑐)𝑎 ↔ ((ω CNF 𝑐)‘𝑓) = 𝐴))
2322reubidva 3396 . . . . . . . . 9 ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑎 = 𝐴) → (∃!𝑓 ∈ dom (ω CNF 𝑐)𝑓(ω CNF 𝑐)𝑎 ↔ ∃!𝑓 ∈ dom (ω CNF 𝑐)((ω CNF 𝑐)‘𝑓) = 𝐴))
24 simpl2 1193 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → 𝑏 ∈ On)
2513a1i 11 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → ω ∈ On)
2624, 15, 253jca 1129 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → (𝑏 ∈ On ∧ 𝑐 ∈ On ∧ ω ∈ On))
27 peano1 7910 . . . . . . . . . . . . 13 ∅ ∈ ω
2826, 27jctir 520 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → ((𝑏 ∈ On ∧ 𝑐 ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω))
29 simprr 773 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → 𝑏𝑐)
30 oewordi 8629 . . . . . . . . . . . 12 (((𝑏 ∈ On ∧ 𝑐 ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) → (𝑏𝑐 → (ω ↑o 𝑏) ⊆ (ω ↑o 𝑐)))
3128, 29, 30sylc 65 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → (ω ↑o 𝑏) ⊆ (ω ↑o 𝑐))
32 simpl3 1194 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → 𝐴 ∈ (ω ↑o 𝑏))
3331, 32sseldd 3984 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → 𝐴 ∈ (ω ↑o 𝑐))
3412, 25, 15cantnff1o 9736 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → (ω CNF 𝑐):dom (ω CNF 𝑐)–1-1-onto→(ω ↑o 𝑐))
35 dff1o5 6857 . . . . . . . . . . . 12 ((ω CNF 𝑐):dom (ω CNF 𝑐)–1-1-onto→(ω ↑o 𝑐) ↔ ((ω CNF 𝑐):dom (ω CNF 𝑐)–1-1→(ω ↑o 𝑐) ∧ ran (ω CNF 𝑐) = (ω ↑o 𝑐)))
36 simpr 484 . . . . . . . . . . . 12 (((ω CNF 𝑐):dom (ω CNF 𝑐)–1-1→(ω ↑o 𝑐) ∧ ran (ω CNF 𝑐) = (ω ↑o 𝑐)) → ran (ω CNF 𝑐) = (ω ↑o 𝑐))
3735, 36sylbi 217 . . . . . . . . . . 11 ((ω CNF 𝑐):dom (ω CNF 𝑐)–1-1-onto→(ω ↑o 𝑐) → ran (ω CNF 𝑐) = (ω ↑o 𝑐))
3834, 37syl 17 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → ran (ω CNF 𝑐) = (ω ↑o 𝑐))
3933, 38eleqtrrd 2844 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → 𝐴 ∈ ran (ω CNF 𝑐))
40 dff1o2 6853 . . . . . . . . . . . 12 ((ω CNF 𝑐):dom (ω CNF 𝑐)–1-1-onto→(ω ↑o 𝑐) ↔ ((ω CNF 𝑐) Fn dom (ω CNF 𝑐) ∧ Fun (ω CNF 𝑐) ∧ ran (ω CNF 𝑐) = (ω ↑o 𝑐)))
41 simp2 1138 . . . . . . . . . . . 12 (((ω CNF 𝑐) Fn dom (ω CNF 𝑐) ∧ Fun (ω CNF 𝑐) ∧ ran (ω CNF 𝑐) = (ω ↑o 𝑐)) → Fun (ω CNF 𝑐))
4240, 41sylbi 217 . . . . . . . . . . 11 ((ω CNF 𝑐):dom (ω CNF 𝑐)–1-1-onto→(ω ↑o 𝑐) → Fun (ω CNF 𝑐))
4334, 42syl 17 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → Fun (ω CNF 𝑐))
44 funcnv3 6636 . . . . . . . . . 10 (Fun (ω CNF 𝑐) ↔ ∀𝑎 ∈ ran (ω CNF 𝑐)∃!𝑓 ∈ dom (ω CNF 𝑐)𝑓(ω CNF 𝑐)𝑎)
4543, 44sylib 218 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → ∀𝑎 ∈ ran (ω CNF 𝑐)∃!𝑓 ∈ dom (ω CNF 𝑐)𝑓(ω CNF 𝑐)𝑎)
4623, 39, 45rspcdv2 3617 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → ∃!𝑓 ∈ dom (ω CNF 𝑐)((ω CNF 𝑐)‘𝑓) = 𝐴)
4732ad2antrr 726 . . . . . . . . . . . . 13 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝐴 ∈ (ω ↑o 𝑏))
48 simplr 769 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝑓 ∈ dom (ω CNF 𝑐))
4913a1i 11 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ω ∈ On)
5015ad2antrr 726 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝑐 ∈ On)
5112, 49, 50cantnfs 9706 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝑓 ∈ dom (ω CNF 𝑐) ↔ (𝑓:𝑐⟶ω ∧ 𝑓 finSupp ∅)))
5248, 51mpbid 232 . . . . . . . . . . . . . 14 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝑓:𝑐⟶ω ∧ 𝑓 finSupp ∅))
53 simpr 484 . . . . . . . . . . . . . 14 ((𝑓:𝑐⟶ω ∧ 𝑓 finSupp ∅) → 𝑓 finSupp ∅)
5452, 53syl 17 . . . . . . . . . . . . 13 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝑓 finSupp ∅)
55 eqid 2737 . . . . . . . . . . . . . . . 16 dom (ω CNF 𝑏) = dom (ω CNF 𝑏)
5624ad2antrr 726 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝑏 ∈ On)
5729ad2antrr 726 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝑏𝑐)
58 simpr 484 . . . . . . . . . . . . . . . . . . 19 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((ω CNF 𝑐)‘𝑓) = 𝐴)
5958, 47eqeltrd 2841 . . . . . . . . . . . . . . . . . 18 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((ω CNF 𝑐)‘𝑓) ∈ (ω ↑o 𝑏))
60 1onn 8678 . . . . . . . . . . . . . . . . . . . . 21 1o ∈ ω
61 ondif2 8540 . . . . . . . . . . . . . . . . . . . . 21 (ω ∈ (On ∖ 2o) ↔ (ω ∈ On ∧ 1o ∈ ω))
6213, 60, 61mpbir2an 711 . . . . . . . . . . . . . . . . . . . 20 ω ∈ (On ∖ 2o)
6362a1i 11 . . . . . . . . . . . . . . . . . . 19 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ω ∈ (On ∖ 2o))
64 cantnfresb 43337 . . . . . . . . . . . . . . . . . . 19 (((ω ∈ (On ∖ 2o) ∧ 𝑐 ∈ On) ∧ (𝑏 ∈ On ∧ 𝑓 ∈ dom (ω CNF 𝑐))) → (((ω CNF 𝑐)‘𝑓) ∈ (ω ↑o 𝑏) ↔ ∀𝑑 ∈ (𝑐𝑏)(𝑓𝑑) = ∅))
6563, 50, 56, 48, 64syl22anc 839 . . . . . . . . . . . . . . . . . 18 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (((ω CNF 𝑐)‘𝑓) ∈ (ω ↑o 𝑏) ↔ ∀𝑑 ∈ (𝑐𝑏)(𝑓𝑑) = ∅))
6659, 65mpbid 232 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ∀𝑑 ∈ (𝑐𝑏)(𝑓𝑑) = ∅)
6766r19.21bi 3251 . . . . . . . . . . . . . . . 16 ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑 ∈ (𝑐𝑏)) → (𝑓𝑑) = ∅)
6827a1i 11 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ∅ ∈ ω)
69 simpllr 776 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑𝑏) → 𝑓 ∈ dom (ω CNF 𝑐))
7013a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑𝑏) → ω ∈ On)
7115adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → 𝑐 ∈ On)
7271ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑𝑏) → 𝑐 ∈ On)
7312, 70, 72cantnfs 9706 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑𝑏) → (𝑓 ∈ dom (ω CNF 𝑐) ↔ (𝑓:𝑐⟶ω ∧ 𝑓 finSupp ∅)))
7469, 73mpbid 232 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑𝑏) → (𝑓:𝑐⟶ω ∧ 𝑓 finSupp ∅))
7574simpld 494 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑𝑏) → 𝑓:𝑐⟶ω)
7657sselda 3983 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑𝑏) → 𝑑𝑐)
7775, 76ffvelcdmd 7105 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑𝑏) → (𝑓𝑑) ∈ ω)
7877fmpttd 7135 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝑑𝑏 ↦ (𝑓𝑑)):𝑏⟶ω)
7912, 25, 15cantnfs 9706 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → (𝑓 ∈ dom (ω CNF 𝑐) ↔ (𝑓:𝑐⟶ω ∧ 𝑓 finSupp ∅)))
8079simprbda 498 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → 𝑓:𝑐⟶ω)
8180adantr 480 . . . . . . . . . . . . . . . . . . 19 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝑓:𝑐⟶ω)
8281, 57feqresmpt 6978 . . . . . . . . . . . . . . . . . 18 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝑓𝑏) = (𝑑𝑏 ↦ (𝑓𝑑)))
8354, 68fsuppres 9433 . . . . . . . . . . . . . . . . . 18 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝑓𝑏) finSupp ∅)
8482, 83eqbrtrrd 5167 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝑑𝑏 ↦ (𝑓𝑑)) finSupp ∅)
8555, 49, 56cantnfs 9706 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((𝑑𝑏 ↦ (𝑓𝑑)) ∈ dom (ω CNF 𝑏) ↔ ((𝑑𝑏 ↦ (𝑓𝑑)):𝑏⟶ω ∧ (𝑑𝑏 ↦ (𝑓𝑑)) finSupp ∅)))
8678, 84, 85mpbir2and 713 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝑑𝑏 ↦ (𝑓𝑑)) ∈ dom (ω CNF 𝑏))
8755, 49, 56, 50, 57, 67, 68, 12, 86cantnfres 9717 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((ω CNF 𝑏)‘(𝑑𝑏 ↦ (𝑓𝑑))) = ((ω CNF 𝑐)‘(𝑑𝑐 ↦ (𝑓𝑑))))
8882fveq2d 6910 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((ω CNF 𝑏)‘(𝑓𝑏)) = ((ω CNF 𝑏)‘(𝑑𝑏 ↦ (𝑓𝑑))))
8981feqmptd 6977 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝑓 = (𝑑𝑐 ↦ (𝑓𝑑)))
9089fveq2d 6910 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((ω CNF 𝑐)‘𝑓) = ((ω CNF 𝑐)‘(𝑑𝑐 ↦ (𝑓𝑑))))
9187, 88, 903eqtr4d 2787 . . . . . . . . . . . . . 14 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((ω CNF 𝑏)‘(𝑓𝑏)) = ((ω CNF 𝑐)‘𝑓))
9291, 58eqtrd 2777 . . . . . . . . . . . . 13 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴)
9347, 54, 923jca 1129 . . . . . . . . . . . 12 (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅ ∧ ((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴))
9493ex 412 . . . . . . . . . . 11 ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (((ω CNF 𝑐)‘𝑓) = 𝐴 → (𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅ ∧ ((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴)))
9594pm4.71rd 562 . . . . . . . . . 10 ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (((ω CNF 𝑐)‘𝑓) = 𝐴 ↔ ((𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅ ∧ ((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))
96 3an4anass 1105 . . . . . . . . . 10 (((𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅ ∧ ((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ↔ ((𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))
9795, 96bitrdi 287 . . . . . . . . 9 ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (((ω CNF 𝑐)‘𝑓) = 𝐴 ↔ ((𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴))))
9897reubidva 3396 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → (∃!𝑓 ∈ dom (ω CNF 𝑐)((ω CNF 𝑐)‘𝑓) = 𝐴 ↔ ∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴))))
9946, 98mpbid 232 . . . . . . 7 (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏𝑐)) → ∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))
10099ex 412 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) → ((𝑐 ∈ On ∧ 𝑏𝑐) → ∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴))))
1019, 100sylbid 240 . . . . 5 ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) → (𝑐 ∈ (On ∖ 𝑏) → ∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴))))
102101ralrimiv 3145 . . . 4 ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) → ∀𝑐 ∈ (On ∖ 𝑏)∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))
1031023exp 1120 . . 3 (𝐴 ∈ On → (𝑏 ∈ On → (𝐴 ∈ (ω ↑o 𝑏) → ∀𝑐 ∈ (On ∖ 𝑏)∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))))
104103reximdvai 3165 . 2 (𝐴 ∈ On → (∃𝑏 ∈ On 𝐴 ∈ (ω ↑o 𝑏) → ∃𝑏 ∈ On ∀𝑐 ∈ (On ∖ 𝑏)∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴))))
1051, 104mpd 15 1 (𝐴 ∈ On → ∃𝑏 ∈ On ∀𝑐 ∈ (On ∖ 𝑏)∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  ∃!wreu 3378  cdif 3948  wss 3951  c0 4333   class class class wbr 5143  cmpt 5225  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  Oncon0 6384  Fun wfun 6555   Fn wfn 6556  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  ωcom 7887  1oc1o 8499  2oc2o 8500  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  ax-inf2 9681
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-ot 4635  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: (None)
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