Theorem cantnf2 41846

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 41764	. 2 ⊢ (𝐴 ∈ On → ∃𝑏 ∈ On 𝐴 ∈ (ω ↑o 𝑏))
2		eldif 3954	. . . . . . 7 ⊢ (𝑐 ∈ (On ∖ 𝑏) ↔ (𝑐 ∈ On ∧ ¬ 𝑐 ∈ 𝑏))
3		simp2 1137	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) → 𝑏 ∈ On)
4		pm3.2 470	. . . . . . . . . 10 ⊢ (𝑏 ∈ On → (𝑐 ∈ On → (𝑏 ∈ On ∧ 𝑐 ∈ On)))
5	3, 4	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) → (𝑐 ∈ On → (𝑏 ∈ On ∧ 𝑐 ∈ On)))
6		ontri1 6387	. . . . . . . . 9 ⊢ ((𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑏 ⊆ 𝑐 ↔ ¬ 𝑐 ∈ 𝑏))
7	5, 6	syl6 35	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) → (𝑐 ∈ On → (𝑏 ⊆ 𝑐 ↔ ¬ 𝑐 ∈ 𝑏)))
8	7	pm5.32d 577	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) → ((𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐) ↔ (𝑐 ∈ On ∧ ¬ 𝑐 ∈ 𝑏)))
9	2, 8	bitr4id 289	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) → (𝑐 ∈ (On ∖ 𝑏) ↔ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)))
10		simplr 767	. . . . . . . . . . . 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 5153	. . . . . . . . . . 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 2731	. . . . . . . . . . . . . 14 ⊢ dom (ω CNF 𝑐) = dom (ω CNF 𝑐)
13		omelon 9623	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝑐 ∈ On)
16	15	ad2antrr 724	. . . . . . . . . . . . . 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 9673	. . . . . . . . . . . . 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 6822	. . . . . . . . . . . . 13 ⊢ ((ω CNF 𝑐):dom (ω CNF 𝑐)–1-1-onto→(ω ↑o 𝑐) → Fun (ω CNF 𝑐))
19	17, 18	syl 17	. . . . . . . . . . . 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 6933	. . . . . . . . . . . 12 ⊢ ((Fun (ω CNF 𝑐) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (((ω CNF 𝑐)‘𝑓) = 𝐴 ↔ 𝑓(ω CNF 𝑐)𝐴))
21	19, 20	sylancom 588	. . . . . . . . . . 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 281	. . . . . . . . . 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 3391	. . . . . . . . 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 1192	. . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → (𝑏 ∈ On ∧ 𝑐 ∈ On ∧ ω ∈ On))
27		peano1 7861	. . . . . . . . . . . . 13 ⊢ ∅ ∈ ω
28	26, 27	jctir 521	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → ((𝑏 ∈ On ∧ 𝑐 ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω))
29		simprr 771	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝑏 ⊆ 𝑐)
30		oewordi 8574	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ On ∧ 𝑐 ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) → (𝑏 ⊆ 𝑐 → (ω ↑o 𝑏) ⊆ (ω ↑o 𝑐)))
31	28, 29, 30	sylc 65	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → (ω ↑o 𝑏) ⊆ (ω ↑o 𝑐))
32		simpl3 1193	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝐴 ∈ (ω ↑o 𝑏))
33	31, 32	sseldd 3979	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝐴 ∈ (ω ↑o 𝑐))
34	12, 25, 15	cantnff1o 9673	. . . . . . . . . . 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 6829	. . . . . . . . . . . 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 485	. . . . . . . . . . . 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 216	. . . . . . . . . . 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 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → ran (ω CNF 𝑐) = (ω ↑o 𝑐))
39	33, 38	eleqtrrd 2835	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝐴 ∈ ran (ω CNF 𝑐))
40		dff1o2 6825	. . . . . . . . . . . 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 1137	. . . . . . . . . . . 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 216	. . . . . . . . . . 11 ⊢ ((ω CNF 𝑐):dom (ω CNF 𝑐)–1-1-onto→(ω ↑o 𝑐) → Fun ◡(ω CNF 𝑐))
43	34, 42	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → Fun ◡(ω CNF 𝑐))
44		funcnv3 6607	. . . . . . . . . 10 ⊢ (Fun ◡(ω CNF 𝑐) ↔ ∀𝑎 ∈ ran (ω CNF 𝑐)∃!𝑓 ∈ dom (ω CNF 𝑐)𝑓(ω CNF 𝑐)𝑎)
45	43, 44	sylib 217	. . . . . . . . 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 3604	. . . . . . . 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 724	. . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . 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 9643	. . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝑐⟶ω ∧ 𝑓 finSupp ∅) → 𝑓 finSupp ∅)
54	52, 53	syl 17	. . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . 16 ⊢ dom (ω CNF 𝑏) = dom (ω CNF 𝑏)
56	24	ad2antrr 724	. . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . 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 2832	. . . . . . . . . . . . . . . . . 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 8622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1o ∈ ω
61		ondif2 8484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ω ∈ (On ∖ 2o) ↔ (ω ∈ On ∧ 1o ∈ ω))
62	13, 60, 61	mpbir2an 709	. . . . . . . . . . . . . . . . . . . 20 ⊢ ω ∈ (On ∖ 2o)
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 ∖ 2o))
64		cantnfresb 41845	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ω ∈ (On ∖ 2o) ∧ 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 837	. . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . 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 3247	. . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → 𝑐 ∈ On)
72	71	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 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 9643	. . . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . 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 3978	. . . . . . . . . . . . . . . . . . 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 7072	. . . . . . . . . . . . . . . . . 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 7099	. . . . . . . . . . . . . . . . 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 9643	. . . . . . . . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → 𝑓:𝑐⟶ω)
81	80	adantr 481	. . . . . . . . . . . . . . . . . . 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 6947	. . . . . . . . . . . . . . . . . 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 9371	. . . . . . . . . . . . . . . . . 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 5165	. . . . . . . . . . . . . . . . 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 9643	. . . . . . . . . . . . . . . . 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 711	. . . . . . . . . . . . . . . 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 9654	. . . . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . 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 6946	. . . . . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 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 1128	. . . . . . . . . . . 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 413	. . . . . . . . . . 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 563	. . . . . . . . . 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 1105	. . . . . . . . . 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 286	. . . . . . . . 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 3391	. . . . . . . 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 231	. . . . . . 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 413	. . . . . 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 239	. . . . 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 3144	. . . 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 1119	. . 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 3164	. 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 15	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)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069  ∃!wreu 3373   ∖ cdif 3941   ⊆ wss 3944  ∅c0 4318   class class class wbr 5141   ↦ cmpt 5224  ^◡ccnv 5668  dom cdm 5669  ran crn 5670   ↾ cres 5671  Oncon0 6353  Fun wfun 6526   Fn wfn 6527  ⟶wf 6528  –1-1→wf1 6529  –1-1-onto→wf1o 6531  ‘cfv 6532  (class class class)co 7393  ωcom 7838  1_oc1o 8441  2_oc2o 8442   ↑_o coe 8447   finSupp cfsupp 9344   CNF ccnf 9638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-inf2 9618

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-ot 4631  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-supp 8129  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-seqom 8430  df-1o 8448  df-2o 8449  df-oadd 8452  df-omul 8453  df-oexp 8454  df-er 8686  df-map 8805  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-fsupp 9345  df-oi 9487  df-cnf 9639

This theorem is referenced by: (None)