Theorem fidcenumlemr 7154

Description: Lemma for fidcenum 7155. Reverse direction (put into deduction form). (Contributed by Jim Kingdon, 19-Oct-2022.)

Hypotheses

Ref	Expression
fidcenumlemr.dc	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
fidcenumlemr.f	⊢ (𝜑 → 𝐹:𝑁–onto→𝐴)
fidcenumlemr.n	⊢ (𝜑 → 𝑁 ∈ ω)

Assertion

Ref	Expression
fidcenumlemr	⊢ (𝜑 → 𝐴 ∈ Fin)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐹,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑁(𝑥,𝑦)

Proof of Theorem fidcenumlemr

Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fidcenumlemr.f	. . 3 ⊢ (𝜑 → 𝐹:𝑁–onto→𝐴)
2		foima 5564	. . 3 ⊢ (𝐹:𝑁–onto→𝐴 → (𝐹 “ 𝑁) = 𝐴)
3	1, 2	syl 14	. 2 ⊢ (𝜑 → (𝐹 “ 𝑁) = 𝐴)
4		ssid 3247	. . 3 ⊢ 𝑁 ⊆ 𝑁
5		fidcenumlemr.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ω)
6	5	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ⊆ 𝑁) → 𝑁 ∈ ω)
7		sseq1 3250	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝑁 ↔ ∅ ⊆ 𝑁))
8	7	anbi2d 464	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ ∅ ⊆ 𝑁)))
9		imaeq2 5072	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝐹 “ 𝑤) = (𝐹 “ ∅))
10	9	eleq1d 2300	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝐹 “ 𝑤) ∈ Fin ↔ (𝐹 “ ∅) ∈ Fin))
11	8, 10	imbi12d 234	. . . . 5 ⊢ (𝑤 = ∅ → (((𝜑 ∧ 𝑤 ⊆ 𝑁) → (𝐹 “ 𝑤) ∈ Fin) ↔ ((𝜑 ∧ ∅ ⊆ 𝑁) → (𝐹 “ ∅) ∈ Fin)))
12		sseq1 3250	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (𝑤 ⊆ 𝑁 ↔ 𝑘 ⊆ 𝑁))
13	12	anbi2d 464	. . . . . 6 ⊢ (𝑤 = 𝑘 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ 𝑘 ⊆ 𝑁)))
14		imaeq2 5072	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (𝐹 “ 𝑤) = (𝐹 “ 𝑘))
15	14	eleq1d 2300	. . . . . 6 ⊢ (𝑤 = 𝑘 → ((𝐹 “ 𝑤) ∈ Fin ↔ (𝐹 “ 𝑘) ∈ Fin))
16	13, 15	imbi12d 234	. . . . 5 ⊢ (𝑤 = 𝑘 → (((𝜑 ∧ 𝑤 ⊆ 𝑁) → (𝐹 “ 𝑤) ∈ Fin) ↔ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)))
17		sseq1 3250	. . . . . . 7 ⊢ (𝑤 = suc 𝑘 → (𝑤 ⊆ 𝑁 ↔ suc 𝑘 ⊆ 𝑁))
18	17	anbi2d 464	. . . . . 6 ⊢ (𝑤 = suc 𝑘 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)))
19		imaeq2 5072	. . . . . . 7 ⊢ (𝑤 = suc 𝑘 → (𝐹 “ 𝑤) = (𝐹 “ suc 𝑘))
20	19	eleq1d 2300	. . . . . 6 ⊢ (𝑤 = suc 𝑘 → ((𝐹 “ 𝑤) ∈ Fin ↔ (𝐹 “ suc 𝑘) ∈ Fin))
21	18, 20	imbi12d 234	. . . . 5 ⊢ (𝑤 = suc 𝑘 → (((𝜑 ∧ 𝑤 ⊆ 𝑁) → (𝐹 “ 𝑤) ∈ Fin) ↔ ((𝜑 ∧ suc 𝑘 ⊆ 𝑁) → (𝐹 “ suc 𝑘) ∈ Fin)))
22		sseq1 3250	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑤 ⊆ 𝑁 ↔ 𝑁 ⊆ 𝑁))
23	22	anbi2d 464	. . . . . 6 ⊢ (𝑤 = 𝑁 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ 𝑁 ⊆ 𝑁)))
24		imaeq2 5072	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝐹 “ 𝑤) = (𝐹 “ 𝑁))
25	24	eleq1d 2300	. . . . . 6 ⊢ (𝑤 = 𝑁 → ((𝐹 “ 𝑤) ∈ Fin ↔ (𝐹 “ 𝑁) ∈ Fin))
26	23, 25	imbi12d 234	. . . . 5 ⊢ (𝑤 = 𝑁 → (((𝜑 ∧ 𝑤 ⊆ 𝑁) → (𝐹 “ 𝑤) ∈ Fin) ↔ ((𝜑 ∧ 𝑁 ⊆ 𝑁) → (𝐹 “ 𝑁) ∈ Fin)))
27		ima0 5095	. . . . . . 7 ⊢ (𝐹 “ ∅) = ∅
28		0fi 7073	. . . . . . 7 ⊢ ∅ ∈ Fin
29	27, 28	eqeltri 2304	. . . . . 6 ⊢ (𝐹 “ ∅) ∈ Fin
30	29	a1i 9	. . . . 5 ⊢ ((𝜑 ∧ ∅ ⊆ 𝑁) → (𝐹 “ ∅) ∈ Fin)
31		simprl 531	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → 𝜑)
32		fofn 5561	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑁–onto→𝐴 → 𝐹 Fn 𝑁)
33	1, 32	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 Fn 𝑁)
34	31, 33	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → 𝐹 Fn 𝑁)
35		simprr 533	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → suc 𝑘 ⊆ 𝑁)
36		vex 2805	. . . . . . . . . . . . . . . 16 ⊢ 𝑘 ∈ V
37	36	sucid 4514	. . . . . . . . . . . . . . 15 ⊢ 𝑘 ∈ suc 𝑘
38	37	a1i 9	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → 𝑘 ∈ suc 𝑘)
39	35, 38	sseldd 3228	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → 𝑘 ∈ 𝑁)
40		fnsnfv 5705	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝑁 ∧ 𝑘 ∈ 𝑁) → {(𝐹‘𝑘)} = (𝐹 “ {𝑘}))
41	34, 39, 40	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → {(𝐹‘𝑘)} = (𝐹 “ {𝑘}))
42	41	uneq2d 3361	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → ((𝐹 “ 𝑘) ∪ {(𝐹‘𝑘)}) = ((𝐹 “ 𝑘) ∪ (𝐹 “ {𝑘})))
43		df-suc 4468	. . . . . . . . . . . . 13 ⊢ suc 𝑘 = (𝑘 ∪ {𝑘})
44	43	imaeq2i 5074	. . . . . . . . . . . 12 ⊢ (𝐹 “ suc 𝑘) = (𝐹 “ (𝑘 ∪ {𝑘}))
45		imaundi 5149	. . . . . . . . . . . 12 ⊢ (𝐹 “ (𝑘 ∪ {𝑘})) = ((𝐹 “ 𝑘) ∪ (𝐹 “ {𝑘}))
46	44, 45	eqtri 2252	. . . . . . . . . . 11 ⊢ (𝐹 “ suc 𝑘) = ((𝐹 “ 𝑘) ∪ (𝐹 “ {𝑘}))
47	42, 46	eqtr4di 2282	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → ((𝐹 “ 𝑘) ∪ {(𝐹‘𝑘)}) = (𝐹 “ suc 𝑘))
48	47	adantr 276	. . . . . . . . 9 ⊢ ((((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) ∧ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → ((𝐹 “ 𝑘) ∪ {(𝐹‘𝑘)}) = (𝐹 “ suc 𝑘))
49		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) ∧ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → (𝐹‘𝑘) ∈ (𝐹 “ 𝑘))
50	49	snssd 3818	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) ∧ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → {(𝐹‘𝑘)} ⊆ (𝐹 “ 𝑘))
51		ssequn2 3380	. . . . . . . . . 10 ⊢ ({(𝐹‘𝑘)} ⊆ (𝐹 “ 𝑘) ↔ ((𝐹 “ 𝑘) ∪ {(𝐹‘𝑘)}) = (𝐹 “ 𝑘))
52	50, 51	sylib 122	. . . . . . . . 9 ⊢ ((((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) ∧ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → ((𝐹 “ 𝑘) ∪ {(𝐹‘𝑘)}) = (𝐹 “ 𝑘))
53	48, 52	eqtr3d 2266	. . . . . . . 8 ⊢ ((((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) ∧ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → (𝐹 “ suc 𝑘) = (𝐹 “ 𝑘))
54		sssucid 4512	. . . . . . . . . . . 12 ⊢ 𝑘 ⊆ suc 𝑘
55		sstr 3235	. . . . . . . . . . . 12 ⊢ ((𝑘 ⊆ suc 𝑘 ∧ suc 𝑘 ⊆ 𝑁) → 𝑘 ⊆ 𝑁)
56	54, 55	mpan 424	. . . . . . . . . . 11 ⊢ (suc 𝑘 ⊆ 𝑁 → 𝑘 ⊆ 𝑁)
57	56	ad2antll 491	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → 𝑘 ⊆ 𝑁)
58		simplr 529	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin))
59	31, 57, 58	mp2and 433	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → (𝐹 “ 𝑘) ∈ Fin)
60	59	adantr 276	. . . . . . . 8 ⊢ ((((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) ∧ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → (𝐹 “ 𝑘) ∈ Fin)
61	53, 60	eqeltrd 2308	. . . . . . 7 ⊢ ((((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) ∧ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → (𝐹 “ suc 𝑘) ∈ Fin)
62	47	adantr 276	. . . . . . . 8 ⊢ ((((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → ((𝐹 “ 𝑘) ∪ {(𝐹‘𝑘)}) = (𝐹 “ suc 𝑘))
63	59	adantr 276	. . . . . . . . 9 ⊢ ((((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → (𝐹 “ 𝑘) ∈ Fin)
64	31, 1	syl 14	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → 𝐹:𝑁–onto→𝐴)
65		fof 5559	. . . . . . . . . . . 12 ⊢ (𝐹:𝑁–onto→𝐴 → 𝐹:𝑁⟶𝐴)
66	64, 65	syl 14	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → 𝐹:𝑁⟶𝐴)
67	66, 39	ffvelcdmd 5783	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → (𝐹‘𝑘) ∈ 𝐴)
68	67	adantr 276	. . . . . . . . 9 ⊢ ((((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → (𝐹‘𝑘) ∈ 𝐴)
69		simpr 110	. . . . . . . . 9 ⊢ ((((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘))
70		unsnfi 7111	. . . . . . . . 9 ⊢ (((𝐹 “ 𝑘) ∈ Fin ∧ (𝐹‘𝑘) ∈ 𝐴 ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → ((𝐹 “ 𝑘) ∪ {(𝐹‘𝑘)}) ∈ Fin)
71	63, 68, 69, 70	syl3anc 1273	. . . . . . . 8 ⊢ ((((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → ((𝐹 “ 𝑘) ∪ {(𝐹‘𝑘)}) ∈ Fin)
72	62, 71	eqeltrrd 2309	. . . . . . 7 ⊢ ((((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → (𝐹 “ suc 𝑘) ∈ Fin)
73		fidcenumlemr.dc	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
74	31, 73	syl 14	. . . . . . . 8 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
75		simpll 527	. . . . . . . 8 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → 𝑘 ∈ ω)
76	74, 64, 75, 57, 67	fidcenumlemrk 7153	. . . . . . 7 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → ((𝐹‘𝑘) ∈ (𝐹 “ 𝑘) ∨ ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)))
77	61, 72, 76	mpjaodan 805	. . . . . 6 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → (𝐹 “ suc 𝑘) ∈ Fin)
78	77	exp31 364	. . . . 5 ⊢ (𝑘 ∈ ω → (((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin) → ((𝜑 ∧ suc 𝑘 ⊆ 𝑁) → (𝐹 “ suc 𝑘) ∈ Fin)))
79	11, 16, 21, 26, 30, 78	finds 4698	. . . 4 ⊢ (𝑁 ∈ ω → ((𝜑 ∧ 𝑁 ⊆ 𝑁) → (𝐹 “ 𝑁) ∈ Fin))
80	6, 79	mpcom 36	. . 3 ⊢ ((𝜑 ∧ 𝑁 ⊆ 𝑁) → (𝐹 “ 𝑁) ∈ Fin)
81	4, 80	mpan2 425	. 2 ⊢ (𝜑 → (𝐹 “ 𝑁) ∈ Fin)
82	3, 81	eqeltrrd 2309	1 ⊢ (𝜑 → 𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  DECID wdc 841   = wceq 1397   ∈ wcel 2202  ∀wral 2510   ∪ cun 3198   ⊆ wss 3200  ∅c0 3494  {csn 3669  suc csuc 4462  ωcom 4688   “ cima 4728   Fn wfn 5321  ⟶wf 5322  –onto→wfo 5324  ‘cfv 5326  Fincfn 6909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6582  df-er 6702  df-en 6910  df-fin 6912

This theorem is referenced by:  fidcenum  7155