Theorem fidcenumlemr 7090

Description: Lemma for fidcenum 7091. 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 5529	. . 3 ⊢ (𝐹:𝑁–onto→𝐴 → (𝐹 “ 𝑁) = 𝐴)
3	1, 2	syl 14	. 2 ⊢ (𝜑 → (𝐹 “ 𝑁) = 𝐴)
4		ssid 3224	. . 3 ⊢ 𝑁 ⊆ 𝑁
5		fidcenumlemr.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ω)
6	5	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ⊆ 𝑁) → 𝑁 ∈ ω)
7		sseq1 3227	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝑁 ↔ ∅ ⊆ 𝑁))
8	7	anbi2d 464	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ ∅ ⊆ 𝑁)))
9		imaeq2 5040	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝐹 “ 𝑤) = (𝐹 “ ∅))
10	9	eleq1d 2278	. . . . . 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 3227	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (𝑤 ⊆ 𝑁 ↔ 𝑘 ⊆ 𝑁))
13	12	anbi2d 464	. . . . . 6 ⊢ (𝑤 = 𝑘 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ 𝑘 ⊆ 𝑁)))
14		imaeq2 5040	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (𝐹 “ 𝑤) = (𝐹 “ 𝑘))
15	14	eleq1d 2278	. . . . . 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 3227	. . . . . . 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 5040	. . . . . . 7 ⊢ (𝑤 = suc 𝑘 → (𝐹 “ 𝑤) = (𝐹 “ suc 𝑘))
20	19	eleq1d 2278	. . . . . 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 3227	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑤 ⊆ 𝑁 ↔ 𝑁 ⊆ 𝑁))
23	22	anbi2d 464	. . . . . 6 ⊢ (𝑤 = 𝑁 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ 𝑁 ⊆ 𝑁)))
24		imaeq2 5040	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝐹 “ 𝑤) = (𝐹 “ 𝑁))
25	24	eleq1d 2278	. . . . . 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 5063	. . . . . . 7 ⊢ (𝐹 “ ∅) = ∅
28		0fin 7014	. . . . . . 7 ⊢ ∅ ∈ Fin
29	27, 28	eqeltri 2282	. . . . . 6 ⊢ (𝐹 “ ∅) ∈ Fin
30	29	a1i 9	. . . . 5 ⊢ ((𝜑 ∧ ∅ ⊆ 𝑁) → (𝐹 “ ∅) ∈ Fin)
31		simprl 529	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → 𝜑)
32		fofn 5526	. . . . . . . . . . . . . . 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 531	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → suc 𝑘 ⊆ 𝑁)
36		vex 2782	. . . . . . . . . . . . . . . 16 ⊢ 𝑘 ∈ V
37	36	sucid 4485	. . . . . . . . . . . . . . 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 3205	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → 𝑘 ∈ 𝑁)
40		fnsnfv 5666	. . . . . . . . . . . . 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 3338	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) → ((𝐹 “ 𝑘) ∪ {(𝐹‘𝑘)}) = ((𝐹 “ 𝑘) ∪ (𝐹 “ {𝑘})))
43		df-suc 4439	. . . . . . . . . . . . 13 ⊢ suc 𝑘 = (𝑘 ∪ {𝑘})
44	43	imaeq2i 5042	. . . . . . . . . . . 12 ⊢ (𝐹 “ suc 𝑘) = (𝐹 “ (𝑘 ∪ {𝑘}))
45		imaundi 5117	. . . . . . . . . . . 12 ⊢ (𝐹 “ (𝑘 ∪ {𝑘})) = ((𝐹 “ 𝑘) ∪ (𝐹 “ {𝑘}))
46	44, 45	eqtri 2230	. . . . . . . . . . 11 ⊢ (𝐹 “ suc 𝑘) = ((𝐹 “ 𝑘) ∪ (𝐹 “ {𝑘}))
47	42, 46	eqtr4di 2260	. . . . . . . . . 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 3792	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) ∧ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → {(𝐹‘𝑘)} ⊆ (𝐹 “ 𝑘))
51		ssequn2 3357	. . . . . . . . . 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 2244	. . . . . . . 8 ⊢ ((((𝑘 ∈ ω ∧ ((𝜑 ∧ 𝑘 ⊆ 𝑁) → (𝐹 “ 𝑘) ∈ Fin)) ∧ (𝜑 ∧ suc 𝑘 ⊆ 𝑁)) ∧ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → (𝐹 “ suc 𝑘) = (𝐹 “ 𝑘))
54		sssucid 4483	. . . . . . . . . . . 12 ⊢ 𝑘 ⊆ suc 𝑘
55		sstr 3212	. . . . . . . . . . . 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 528	. . . . . . . . . 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 2286	. . . . . . 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 5524	. . . . . . . . . . . 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 5744	. . . . . . . . . 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 7049	. . . . . . . . 9 ⊢ (((𝐹 “ 𝑘) ∈ Fin ∧ (𝐹‘𝑘) ∈ 𝐴 ∧ ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑘)) → ((𝐹 “ 𝑘) ∪ {(𝐹‘𝑘)}) ∈ Fin)
71	63, 68, 69, 70	syl3anc 1252	. . . . . . . 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 2287	. . . . . . 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 7089	. . . . . . 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 802	. . . . . 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 4669	. . . 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 2287	1 ⊢ (𝜑 → 𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  DECID wdc 838   = wceq 1375   ∈ wcel 2180  ∀wral 2488   ∪ cun 3175   ⊆ wss 3177  ∅c0 3471  {csn 3646  suc csuc 4433  ωcom 4659   “ cima 4699   Fn wfn 5289  ⟶wf 5290  –onto→wfo 5292  ‘cfv 5294  Fincfn 6857

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-v 2781  df-sbc 3009  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-1o 6532  df-er 6650  df-en 6858  df-fin 6860

This theorem is referenced by:  fidcenum  7091