cantnf2 - Mathbox for Richard Penner

	Mathbox for Richard Penner		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cantnf2		Structured version Visualization version GIF version

Theorem cantnf2 43682

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 43601	. 2 ⊢ (𝐴 ∈ On → ∃𝑏 ∈ On 𝐴 ∈ (ω ↑_o 𝑏))
2		eldif 3913	. . . . . . 7 ⊢ (𝑐 ∈ (On ∖ 𝑏) ↔ (𝑐 ∈ On ∧ ¬ 𝑐 ∈ 𝑏))
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)))
5	3, 4	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) → (𝑐 ∈ On → (𝑏 ∈ On ∧ 𝑐 ∈ On)))
6		ontri1 6359	. . . . . . . . 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 290	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) → (𝑐 ∈ (On ∖ 𝑏) ↔ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)))
10		simplr 769	. . . . . . . . . . . 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 5112	. . . . . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ dom (ω CNF 𝑐) = dom (ω CNF 𝑐)
13		omelon 9567	. . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝑐 ∈ On)
16	15	ad2antrr 727	. . . . . . . . . . . . . 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 9617	. . . . . . . . . . . . 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 6784	. . . . . . . . . . . . 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 6895	. . . . . . . . . . . 12 ⊢ ((Fun (ω CNF 𝑐) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → (((ω CNF 𝑐)‘𝑓) = 𝐴 ↔ 𝑓(ω CNF 𝑐)𝐴))
21	19, 20	sylancom 589	. . . . . . . . . . 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 282	. . . . . . . . . 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 3366	. . . . . . . . 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 1194	. . . . . . . . . . . . . 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 1129	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → (𝑏 ∈ On ∧ 𝑐 ∈ On ∧ ω ∈ On))
27		peano1 7841	. . . . . . . . . . . . 13 ⊢ ∅ ∈ ω
28	26, 27	jctir 520	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → ((𝑏 ∈ On ∧ 𝑐 ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω))
29		simprr 773	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝑏 ⊆ 𝑐)
30		oewordi 8529	. . . . . . . . . . . 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 1195	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝐴 ∈ (ω ↑_o 𝑏))
33	31, 32	sseldd 3936	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝐴 ∈ (ω ↑_o 𝑐))
34	12, 25, 15	cantnff1o 9617	. . . . . . . . . . 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 6791	. . . . . . . . . . . 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))
37	35, 36	sylbi 217	. . . . . . . . . . 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 2840	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) → 𝐴 ∈ ran (ω CNF 𝑐))
40		dff1o2 6787	. . . . . . . . . . . 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))
42	40, 41	sylbi 217	. . . . . . . . . . 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 6570	. . . . . . . . . 10 ⊢ (Fun ^◡(ω CNF 𝑐) ↔ ∀𝑎 ∈ ran (ω CNF 𝑐)∃!𝑓 ∈ dom (ω CNF 𝑐)𝑓(ω CNF 𝑐)𝑎)
45	43, 44	sylib 218	. . . . . . . . 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 3573	. . . . . . . 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 727	. . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . 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 9587	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . 16 ⊢ dom (ω CNF 𝑏) = dom (ω CNF 𝑏)
56	24	ad2antrr 727	. . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . . . 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 8578	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1_o ∈ ω
61		ondif2 8439	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ω ∈ (On ∖ 2_o) ↔ (ω ∈ On ∧ 1_o ∈ ω))
62	13, 60, 61	mpbir2an 712	. . . . . . . . . . . . . . . . . . . 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 43681	. . . . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . 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 3230	. . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → 𝑐 ∈ On)
72	71	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . 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 9587	. . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . . . . 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 3935	. . . . . . . . . . . . . . . . . . 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 7039	. . . . . . . . . . . . . . . . . 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 7069	. . . . . . . . . . . . . . . . 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 9587	. . . . . . . . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ On ∧ 𝑏 ∈ On ∧ 𝐴 ∈ (ω ↑_o 𝑏)) ∧ (𝑐 ∈ On ∧ 𝑏 ⊆ 𝑐)) ∧ 𝑓 ∈ dom (ω CNF 𝑐)) → 𝑓:𝑐⟶ω)
81	80	adantr 480	. . . . . . . . . . . . . . . . . . 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 6911	. . . . . . . . . . . . . . . . . 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 9308	. . . . . . . . . . . . . . . . . 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 5124	. . . . . . . . . . . . . . . . 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 9587	. . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . 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 9598	. . . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . 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 1129	. . . . . . . . . . . 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 412	. . . . . . . . . . 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 562	. . . . . . . . . 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 287	. . . . . . . . 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 3366	. . . . . . . 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 232	. . . . . . 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 412	. . . . . 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 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)‘(𝑓 ↾ 𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴))))
102	101	ralrimiv 3129	. . . 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 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)‘(𝑓 ↾ 𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))))
104	103	reximdvai 3149	. 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)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ∃!wreu 3350 ∖ cdif 3900 ⊆ wss 3903 ∅c0 4287 class class class wbr 5100 ↦ cmpt 5181 ^◡ccnv 5631 dom cdm 5632 ran crn 5633 ↾ cres 5634 Oncon0 6325 Fun wfun 6494 Fn wfn 6495 ⟶wf 6496 –1-1→wf1 6497 –1-1-onto→wf1o 6499 ‘cfv 6500 (class class class)co 7368 ωcom 7818 1_oc1o 8400 2_oc2o 8401 ↑_o coe 8406 finSupp cfsupp 9276 CNF ccnf 9582

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-ot 4591 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-seqom 8389 df-1o 8407 df-2o 8408 df-oadd 8411 df-omul 8412 df-oexp 8413 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-oi 9427 df-cnf 9583

This theorem is referenced by: (None)

W3C validator