cantnf2 - Mathbox for Richard Penner

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Theorem cantnf2 44167

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)‘(𝑓 ↾ 𝑏)) = 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.

Step	Hyp	Ref	Expression
1		onexoegt 44086	. 2 ⊢ (𝐴 ∈ On → ∃𝑏 ∈ On 𝐴 ∈ (ω ↑_o 𝑏))
2		eldif 3909	. . . . . . 7 ⊢ (𝑐 ∈ (On ∖ 𝑏) ↔ (𝑐 ∈ On ∧ ¬ 𝑐 ∈ 𝑏))
3		simp2 1155	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) → 𝑏 ∈ On)
4		pm3.2 475	. . . . . . . . . 10 ⊢ (𝑏 ∈ On → (𝑐 ∈ On → (𝑏 ∈ On ∧ 𝑐 ∈ On)))
5	3, 4	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) → (𝑐 ∈ On → (𝑏 ∈ On ∧ 𝑐 ∈ On)))
6		ontri1 6392	. . . . . . . . 9 ⊢ ((𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑏 ⊆ 𝑐 ↔ ¬ 𝑐 ∈ 𝑏))
7	5, 6	syl6 36	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) → (𝑐 ∈ On → (𝑏 ⊆ 𝑐 ↔ ¬ 𝑐 ∈ 𝑏)))
8	7	pm5.32d 588	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) → ((𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐) ↔ (𝑐 ∈ On ∧ ¬ 𝑐 ∈ 𝑏)))
9	2, 8	bitr4id 293	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) → (𝑐 ∈ (On ∖ 𝑏) ↔ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)))
10		simplr 781	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → 𝑎 = 𝐴)
11	10	breq2d 5115	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (𝑓(ω CNF 𝑐)𝑎 ↔ 𝑓(ω CNF 𝑐)𝐴))
12		eqid 2760	. . . . . . . . . . . . . 14 ⊢ dom (ω CNF 𝑐) = dom (ω CNF 𝑐)
13		omelon 9626	. . . . . . . . . . . . . . 15 ⊢ ω ∈ On
14	13	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → ω ∈ On)
15		simprl 783	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝑐 ∈ On)
16	15	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → 𝑐 ∈ On)
17	12, 14, 16	cantnff1o 9676	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (ω CNF 𝑐):dom (ω CNF 𝑐)–1-1-onto→(ω ↑_o 𝑐))
18		f1ofun 6820	. . . . . . . . . . . . 13 ⊢ ((ω CNF 𝑐):dom (ω CNF 𝑐)–1-1-onto→(ω ↑_o 𝑐) → Fun (ω CNF 𝑐))
19	17, 18	syl 18	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → Fun (ω CNF 𝑐))
20		funbrfvb 6932	. . . . . . . . . . . 12 ⊢ ((Fun (ω CNF 𝑐) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (((ω CNF 𝑐)‘𝑓) = 𝐴 ↔ 𝑓(ω CNF 𝑐)𝐴))
21	19, 20	sylancom 600	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (((ω CNF 𝑐)‘𝑓) = 𝐴 ↔ 𝑓(ω CNF 𝑐)𝐴))
22	11, 21	bitr4d 285	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑎 = 𝐴) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (𝑓(ω CNF 𝑐)𝑎 ↔ ((ω CNF 𝑐)‘𝑓) = 𝐴))
23	22	reubidva 3379	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑎 = 𝐴) → (∃!𝑓 ∈ dom (ω CNF 𝑐)𝑓(ω CNF 𝑐)𝑎 ↔ ∃!𝑓 ∈ dom (ω CNF 𝑐)((ω CNF 𝑐)‘𝑓) = 𝐴))
24		simpl2 1211	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝑏 ∈ On)
25	13	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → ω ∈ On)
26	24, 15, 25	3jca 1146	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → (𝑏 ∈ On ∧ 𝑐 ∈ On ∧ ω ∈ On))
27		peano1 7886	. . . . . . . . . . . . 13 ⊢ ∅ ∈ ω
28	26, 27	jctir 530	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → ((𝑏 ∈ On ∧ 𝑐 ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω))
29		simprr 785	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝑏 ⊆ 𝑐)
30		oewordi 8580	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ On ∧ 𝑐 ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) → (𝑏 ⊆ 𝑐 → (ω ↑_o 𝑏) ⊆ (ω ↑_o 𝑐)))
31	28, 29, 30	sylc 66	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → (ω ↑_o 𝑏) ⊆ (ω ↑_o 𝑐))
32		simpl3 1212	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝐴 ∈ (ω ↑_o 𝑏))
33	31, 32	sseldd 3932	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝐴 ∈ (ω ↑_o 𝑐))
34	12, 25, 15	cantnff1o 9676	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → (ω CNF 𝑐):dom (ω CNF 𝑐)–1-1-onto→(ω ↑_o 𝑐))
35		dff1o5 6828	. . . . . . . . . . . 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 490	. . . . . . . . . . . 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))
37	35, 36	sylbi 220	. . . . . . . . . . 11 ⊢ ((ω CNF 𝑐):dom (ω CNF 𝑐)–1-1-onto→(ω ↑_o 𝑐) → ran (ω CNF 𝑐) = (ω ↑_o 𝑐))
38	34, 37	syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → ran (ω CNF 𝑐) = (ω ↑_o 𝑐))
39	33, 38	eleqtrrd 2863	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝐴 ∈ ran (ω CNF 𝑐))
40		dff1o2 6824	. . . . . . . . . . . 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 1155	. . . . . . . . . . . 12 ⊢ (((ω CNF 𝑐) Fn dom (ω CNF 𝑐) ∧ Fun ^◡(ω CNF 𝑐) ∧ ran (ω CNF 𝑐) = (ω ↑_o 𝑐)) → Fun ^◡(ω CNF 𝑐))
42	40, 41	sylbi 220	. . . . . . . . . . 11 ⊢ ((ω CNF 𝑐):dom (ω CNF 𝑐)–1-1-onto→(ω ↑_o 𝑐) → Fun ^◡(ω CNF 𝑐))
43	34, 42	syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → Fun ^◡(ω CNF 𝑐))
44		funcnv3 6604	. . . . . . . . . 10 ⊢ (Fun ^◡(ω CNF 𝑐) ↔ ∀𝑎 ∈ ran (ω CNF 𝑐)∃!𝑓 ∈ dom (ω CNF 𝑐)𝑓(ω CNF 𝑐)𝑎)
45	43, 44	sylib 221	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → ∀𝑎 ∈ ran (ω CNF 𝑐)∃!𝑓 ∈ dom (ω CNF 𝑐)𝑓(ω CNF 𝑐)𝑎)
46	23, 39, 45	rspcdv2 3571	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → ∃!𝑓 ∈ dom (ω CNF 𝑐)((ω CNF 𝑐)‘𝑓) = 𝐴)
47	32	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝐴 ∈ (ω ↑_o 𝑏))
48		simplr 781	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝑓 ∈ dom (ω CNF 𝑐))
49	13	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ω ∈ On)
50	15	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝑐 ∈ On)
51	12, 49, 50	cantnfs 9646	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝑓 ∈ dom (ω CNF 𝑐) ↔ (𝑓:𝑐⟶ω ∧ 𝑓 finSupp ∅)))
52	48, 51	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝑓:𝑐⟶ω ∧ 𝑓 finSupp ∅))
53		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝑐⟶ω ∧ 𝑓 finSupp ∅) → 𝑓 finSupp ∅)
54	52, 53	syl 18	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝑓 finSupp ∅)
55		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ dom (ω CNF 𝑏) = dom (ω CNF 𝑏)
56	24	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝑏 ∈ On)
57	29	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝑏 ⊆ 𝑐)
58		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((ω CNF 𝑐)‘𝑓) = 𝐴)
59	58, 47	eqeltrd 2860	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((ω CNF 𝑐)‘𝑓) ∈ (ω ↑_o 𝑏))
60		1onn 8629	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1_o ∈ ω
61		ondif2 8490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ω ∈ (On ∖ 2_o) ↔ (ω ∈ On ∧ 1_o ∈ ω))
62	13, 60, 61	mpbir2an 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ ω ∈ (On ∖ 2_o)
63	62	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ω ∈ (On ∖ 2_o))
64		cantnfresb 44166	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ω ∈ (On ∖ 2_o) ∧ 𝑐 ∈ On) ∧ (𝑏 ∈ On ∧ 𝑓 ∈ dom (ω CNF 𝑐))) → (((ω CNF 𝑐)‘𝑓) ∈ (ω ↑_o 𝑏) ↔ ∀𝑑 ∈ (𝑐 ∖ 𝑏)(𝑓‘𝑑) = ∅))
65	63, 50, 56, 48, 64	syl22anc 852	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (((ω CNF 𝑐)‘𝑓) ∈ (ω ↑_o 𝑏) ↔ ∀𝑑 ∈ (𝑐 ∖ 𝑏)(𝑓‘𝑑) = ∅))
66	59, 65	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ∀𝑑 ∈ (𝑐 ∖ 𝑏)(𝑓‘𝑑) = ∅)
67	66	r19.21bi 3254	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑 ∈ (𝑐 ∖ 𝑏)) → (𝑓‘𝑑) = ∅)
68	27	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ∅ ∈ ω)
69		simpllr 788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑 ∈ 𝑏) → 𝑓 ∈ dom (ω CNF 𝑐))
70	13	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑 ∈ 𝑏) → ω ∈ On)
71	15	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → 𝑐 ∈ On)
72	71	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑 ∈ 𝑏) → 𝑐 ∈ On)
73	12, 70, 72	cantnfs 9646	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑 ∈ 𝑏) → (𝑓 ∈ dom (ω CNF 𝑐) ↔ (𝑓:𝑐⟶ω ∧ 𝑓 finSupp ∅)))
74	69, 73	mpbid 235	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑 ∈ 𝑏) → (𝑓:𝑐⟶ω ∧ 𝑓 finSupp ∅))
75	74	simpld 500	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑 ∈ 𝑏) → 𝑓:𝑐⟶ω)
76	57	sselda 3931	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑 ∈ 𝑏) → 𝑑 ∈ 𝑐)
77	75, 76	ffvelcdmd 7079	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ∧ 𝑑 ∈ 𝑏) → (𝑓‘𝑑) ∈ ω)
78	77	fmpttd 7109	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝑑 ∈ 𝑏 ↦ (𝑓‘𝑑)):𝑏⟶ω)
79	12, 25, 15	cantnfs 9646	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → (𝑓 ∈ dom (ω CNF 𝑐) ↔ (𝑓:𝑐⟶ω ∧ 𝑓 finSupp ∅)))
80	79	simprbda 504	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → 𝑓:𝑐⟶ω)
81	80	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝑓:𝑐⟶ω)
82	81, 57	feqresmpt 6948	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝑓 ↾ 𝑏) = (𝑑 ∈ 𝑏 ↦ (𝑓‘𝑑)))
83	54, 68	fsuppres 9364	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝑓 ↾ 𝑏) finSupp ∅)
84	82, 83	eqbrtrrd 5129	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝑑 ∈ 𝑏 ↦ (𝑓‘𝑑)) finSupp ∅)
85	55, 49, 56	cantnfs 9646	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((𝑑 ∈ 𝑏 ↦ (𝑓‘𝑑)) ∈ dom (ω CNF 𝑏) ↔ ((𝑑 ∈ 𝑏 ↦ (𝑓‘𝑑)):𝑏⟶ω ∧ (𝑑 ∈ 𝑏 ↦ (𝑓‘𝑑)) finSupp ∅)))
86	78, 84, 85	mpbir2and 726	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝑑 ∈ 𝑏 ↦ (𝑓‘𝑑)) ∈ dom (ω CNF 𝑏))
87	55, 49, 56, 50, 57, 67, 68, 12, 86	cantnfres 9657	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((ω CNF 𝑏)‘(𝑑 ∈ 𝑏 ↦ (𝑓‘𝑑))) = ((ω CNF 𝑐)‘(𝑑 ∈ 𝑐 ↦ (𝑓‘𝑑))))
88	82	fveq2d 6883	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = ((ω CNF 𝑏)‘(𝑑 ∈ 𝑏 ↦ (𝑓‘𝑑))))
89	81	feqmptd 6947	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → 𝑓 = (𝑑 ∈ 𝑐 ↦ (𝑓‘𝑑)))
90	89	fveq2d 6883	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((ω CNF 𝑐)‘𝑓) = ((ω CNF 𝑐)‘(𝑑 ∈ 𝑐 ↦ (𝑓‘𝑑))))
91	87, 88, 90	3eqtr4d 2805	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = ((ω CNF 𝑐)‘𝑓))
92	91, 58	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → ((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = 𝐴)
93	47, 54, 92	3jca 1146	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) → (𝐴 ∈ (ω ↑_o 𝑏) ∧ 𝑓 finSupp ∅ ∧ ((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = 𝐴))
94	93	ex 418	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (((ω CNF 𝑐)‘𝑓) = 𝐴 → (𝐴 ∈ (ω ↑_o 𝑏) ∧ 𝑓 finSupp ∅ ∧ ((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = 𝐴)))
95	94	pm4.71rd 572	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (((ω CNF 𝑐)‘𝑓) = 𝐴 ↔ ((𝐴 ∈ (ω ↑_o 𝑏) ∧ 𝑓 finSupp ∅ ∧ ((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = 𝐴) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))
96		3an4anass 1122	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ω ↑_o 𝑏) ∧ 𝑓 finSupp ∅ ∧ ((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = 𝐴) ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴) ↔ ((𝐴 ∈ (ω ↑_o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))
97	95, 96	bitrdi 290	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (((ω CNF 𝑐)‘𝑓) = 𝐴 ↔ ((𝐴 ∈ (ω ↑_o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴))))
98	97	reubidva 3379	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → (∃!𝑓 ∈ dom (ω CNF 𝑐)((ω CNF 𝑐)‘𝑓) = 𝐴 ↔ ∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑_o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴))))
99	46, 98	mpbid 235	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → ∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑_o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))
100	99	ex 418	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) → ((𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐) → ∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑_o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴))))
101	9, 100	sylbid 243	. . . . 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)‘(𝑓 ↾ 𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴))))
102	101	ralrimiv 3153	. . . 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)‘(𝑓 ↾ 𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))
103	102	3exp 1137	. . 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)‘(𝑓 ↾ 𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))))
104	103	reximdvai 3173	. 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)‘(𝑓 ↾ 𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴))))
105	1, 104	mpd 16	1 ⊢ (𝐴 ∈ On → ∃𝑏 ∈ On ∀𝑐 ∈ (On ∖ 𝑏)∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑_o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∃wrex 3086 ∃!wreu 3363 ∖ cdif 3896 ⊆ wss 3899 ∅c0 4279 class class class wbr 5103 ↦ cmpt 5186 ^◡ccnv 5654 dom cdm 5655 ran crn 5656 ↾ cres 5657 Oncon0 6357 Fun wfun 6527 Fn wfn 6528 ⟶wf 6529 –1-1→wf1 6530 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7414 ωcom 7863 1_oc1o 8449 2_oc2o 8450 ↑_o coe 8455 finSupp cfsupp 9332 CNF ccnf 9641

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9621

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-seqom 8438 df-1o 8456 df-2o 8457 df-oadd 8460 df-omul 8461 df-oexp 8462 df-er 8697 df-map 8829 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9333 df-oi 9483 df-cnf 9642

This theorem is used by: (None)

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