Theorem finomni 7315

Description: A finite set is omniscient. Remark right after Definition 3.1 of [Pierik], p. 14. (Contributed by Jim Kingdon, 28-Jun-2022.)

Assertion

Ref	Expression
finomni	⊢ (𝐴 ∈ Fin → 𝐴 ∈ Omni)

Proof of Theorem finomni

Dummy variables 𝑤 𝑦 𝑧 𝑓 𝑔 𝑥 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2292	. 2 ⊢ (𝑤 = ∅ → (𝑤 ∈ Omni ↔ ∅ ∈ Omni))
2		eleq1 2292	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ Omni ↔ 𝑦 ∈ Omni))
3		eleq1 2292	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ∈ Omni ↔ (𝑦 ∪ {𝑧}) ∈ Omni))
4		eleq1 2292	. 2 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ Omni ↔ 𝐴 ∈ Omni))
5		0ex 4211	. . . 4 ⊢ ∅ ∈ V
6		isomni 7311	. . . 4 ⊢ (∅ ∈ V → (∅ ∈ Omni ↔ ∀𝑓(𝑓:∅⟶2o → (∃𝑥 ∈ ∅ (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ ∅ (𝑓‘𝑥) = 1o))))
7	5, 6	ax-mp 5	. . 3 ⊢ (∅ ∈ Omni ↔ ∀𝑓(𝑓:∅⟶2o → (∃𝑥 ∈ ∅ (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ ∅ (𝑓‘𝑥) = 1o)))
8		ral0 3593	. . . . 5 ⊢ ∀𝑥 ∈ ∅ (𝑓‘𝑥) = 1o
9	8	olci 737	. . . 4 ⊢ (∃𝑥 ∈ ∅ (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ ∅ (𝑓‘𝑥) = 1o)
10	9	a1i 9	. . 3 ⊢ (𝑓:∅⟶2o → (∃𝑥 ∈ ∅ (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ ∅ (𝑓‘𝑥) = 1o))
11	7, 10	mpgbir 1499	. 2 ⊢ ∅ ∈ Omni
12		elun1 3371	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑦 → 𝑥 ∈ (𝑦 ∪ {𝑧}))
13	12	ad2antlr 489	. . . . . . . . . . 11 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑔 ↾ 𝑦)‘𝑥) = ∅) → 𝑥 ∈ (𝑦 ∪ {𝑧}))
14		fvres 5653	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑦 → ((𝑔 ↾ 𝑦)‘𝑥) = (𝑔‘𝑥))
15	14	ad2antlr 489	. . . . . . . . . . . 12 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑔 ↾ 𝑦)‘𝑥) = ∅) → ((𝑔 ↾ 𝑦)‘𝑥) = (𝑔‘𝑥))
16		simpr 110	. . . . . . . . . . . 12 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑔 ↾ 𝑦)‘𝑥) = ∅) → ((𝑔 ↾ 𝑦)‘𝑥) = ∅)
17	15, 16	eqtr3d 2264	. . . . . . . . . . 11 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑔 ↾ 𝑦)‘𝑥) = ∅) → (𝑔‘𝑥) = ∅)
18		fveq2 5629	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑥 → (𝑔‘𝑢) = (𝑔‘𝑥))
19	18	eqeq1d 2238	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑥 → ((𝑔‘𝑢) = ∅ ↔ (𝑔‘𝑥) = ∅))
20	19	rspcev 2907	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝑦 ∪ {𝑧}) ∧ (𝑔‘𝑥) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅)
21	13, 17, 20	syl2anc 411	. . . . . . . . . 10 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑔 ↾ 𝑦)‘𝑥) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅)
22	21	orcd 738	. . . . . . . . 9 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑔 ↾ 𝑦)‘𝑥) = ∅) → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o))
23	22	ex 115	. . . . . . . 8 ⊢ ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥 ∈ 𝑦) → (((𝑔 ↾ 𝑦)‘𝑥) = ∅ → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o)))
24	23	rexlimdva 2648	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (∃𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = ∅ → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o)))
25		vsnid 3698	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ {𝑧}
26		elun2 3372	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
27	25, 26	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
28	27	a1i 9	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
29		fveq2 5629	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → (𝑔‘𝑢) = (𝑔‘𝑧))
30	29	eqeq1d 2238	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑧 → ((𝑔‘𝑢) = ∅ ↔ (𝑔‘𝑧) = ∅))
31	30	rspcev 2907	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝑦 ∪ {𝑧}) ∧ (𝑔‘𝑧) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅)
32	28, 31	sylan 283	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔‘𝑧) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅)
33	32	orcd 738	. . . . . . . . 9 ⊢ ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔‘𝑧) = ∅) → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o))
34	33	a1d 22	. . . . . . . 8 ⊢ ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔‘𝑧) = ∅) → (∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o)))
35		simpr 110	. . . . . . . . . . . 12 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔‘𝑧) = 1o) ∧ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o) → ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o)
36		fveq2 5629	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → ((𝑔 ↾ 𝑦)‘𝑥) = ((𝑔 ↾ 𝑦)‘𝑢))
37	36	eqeq1d 2238	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (((𝑔 ↾ 𝑦)‘𝑥) = 1o ↔ ((𝑔 ↾ 𝑦)‘𝑢) = 1o))
38	37	cbvralv 2765	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o ↔ ∀𝑢 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑢) = 1o)
39		fvres 5653	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑦 → ((𝑔 ↾ 𝑦)‘𝑢) = (𝑔‘𝑢))
40	39	eqeq1d 2238	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ 𝑦 → (((𝑔 ↾ 𝑦)‘𝑢) = 1o ↔ (𝑔‘𝑢) = 1o))
41	40	ralbiia 2544	. . . . . . . . . . . . 13 ⊢ (∀𝑢 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑢) = 1o ↔ ∀𝑢 ∈ 𝑦 (𝑔‘𝑢) = 1o)
42	38, 41	bitri 184	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o ↔ ∀𝑢 ∈ 𝑦 (𝑔‘𝑢) = 1o)
43	35, 42	sylib 122	. . . . . . . . . . 11 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔‘𝑧) = 1o) ∧ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o) → ∀𝑢 ∈ 𝑦 (𝑔‘𝑢) = 1o)
44		simplr 528	. . . . . . . . . . . 12 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔‘𝑧) = 1o) ∧ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o) → (𝑔‘𝑧) = 1o)
45		vex 2802	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
46	29	eqeq1d 2238	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → ((𝑔‘𝑢) = 1o ↔ (𝑔‘𝑧) = 1o))
47	45, 46	ralsn 3709	. . . . . . . . . . . 12 ⊢ (∀𝑢 ∈ {𝑧} (𝑔‘𝑢) = 1o ↔ (𝑔‘𝑧) = 1o)
48	44, 47	sylibr 134	. . . . . . . . . . 11 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔‘𝑧) = 1o) ∧ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o) → ∀𝑢 ∈ {𝑧} (𝑔‘𝑢) = 1o)
49		ralun 3386	. . . . . . . . . . 11 ⊢ ((∀𝑢 ∈ 𝑦 (𝑔‘𝑢) = 1o ∧ ∀𝑢 ∈ {𝑧} (𝑔‘𝑢) = 1o) → ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o)
50	43, 48, 49	syl2anc 411	. . . . . . . . . 10 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔‘𝑧) = 1o) ∧ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o) → ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o)
51	50	olcd 739	. . . . . . . . 9 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔‘𝑧) = 1o) ∧ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o) → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o))
52	51	ex 115	. . . . . . . 8 ⊢ ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔‘𝑧) = 1o) → (∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o)))
53		simpr 110	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → 𝑔:(𝑦 ∪ {𝑧})⟶2o)
54	53, 28	ffvelcdmd 5773	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (𝑔‘𝑧) ∈ 2o)
55		df2o3 6583	. . . . . . . . . 10 ⊢ 2o = {∅, 1o}
56	54, 55	eleqtrdi 2322	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (𝑔‘𝑧) ∈ {∅, 1o})
57		elpri 3689	. . . . . . . . 9 ⊢ ((𝑔‘𝑧) ∈ {∅, 1o} → ((𝑔‘𝑧) = ∅ ∨ (𝑔‘𝑧) = 1o))
58	56, 57	syl 14	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → ((𝑔‘𝑧) = ∅ ∨ (𝑔‘𝑧) = 1o))
59	34, 52, 58	mpjaodan 803	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o)))
60		vex 2802	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
61		isomni 7311	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → (𝑦 ∈ Omni ↔ ∀𝑓(𝑓:𝑦⟶2o → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o))))
62	60, 61	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑦 ∈ Omni ↔ ∀𝑓(𝑓:𝑦⟶2o → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)))
63	62	biimpi 120	. . . . . . . . . 10 ⊢ (𝑦 ∈ Omni → ∀𝑓(𝑓:𝑦⟶2o → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)))
64	63	adantl 277	. . . . . . . . 9 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → ∀𝑓(𝑓:𝑦⟶2o → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)))
65		vex 2802	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
66	65	resex 5046	. . . . . . . . . 10 ⊢ (𝑔 ↾ 𝑦) ∈ V
67		feq1 5456	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → (𝑓:𝑦⟶2o ↔ (𝑔 ↾ 𝑦):𝑦⟶2o))
68		fveq1 5628	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → (𝑓‘𝑥) = ((𝑔 ↾ 𝑦)‘𝑥))
69	68	eqeq1d 2238	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → ((𝑓‘𝑥) = ∅ ↔ ((𝑔 ↾ 𝑦)‘𝑥) = ∅))
70	69	rexbidv 2531	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = ∅))
71	68	eqeq1d 2238	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → ((𝑓‘𝑥) = 1o ↔ ((𝑔 ↾ 𝑦)‘𝑥) = 1o))
72	71	ralbidv 2530	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → (∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o))
73	70, 72	orbi12d 798	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → ((∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o) ↔ (∃𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o)))
74	67, 73	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → ((𝑓:𝑦⟶2o → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)) ↔ ((𝑔 ↾ 𝑦):𝑦⟶2o → (∃𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o))))
75	66, 74	spcv 2897	. . . . . . . . 9 ⊢ (∀𝑓(𝑓:𝑦⟶2o → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)) → ((𝑔 ↾ 𝑦):𝑦⟶2o → (∃𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o)))
76	64, 75	syl 14	. . . . . . . 8 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → ((𝑔 ↾ 𝑦):𝑦⟶2o → (∃𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o)))
77		ssun1 3367	. . . . . . . . 9 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
78		fssres 5503	. . . . . . . . 9 ⊢ ((𝑔:(𝑦 ∪ {𝑧})⟶2o ∧ 𝑦 ⊆ (𝑦 ∪ {𝑧})) → (𝑔 ↾ 𝑦):𝑦⟶2o)
79	77, 78	mpan2 425	. . . . . . . 8 ⊢ (𝑔:(𝑦 ∪ {𝑧})⟶2o → (𝑔 ↾ 𝑦):𝑦⟶2o)
80	76, 79	impel 280	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (∃𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o))
81	24, 59, 80	mpjaod 723	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o))
82	81	ex 115	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → (𝑔:(𝑦 ∪ {𝑧})⟶2o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o)))
83	82	alrimiv 1920	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → ∀𝑔(𝑔:(𝑦 ∪ {𝑧})⟶2o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o)))
84	45	snex 4269	. . . . . 6 ⊢ {𝑧} ∈ V
85	60, 84	unex 4532	. . . . 5 ⊢ (𝑦 ∪ {𝑧}) ∈ V
86		isomni 7311	. . . . 5 ⊢ ((𝑦 ∪ {𝑧}) ∈ V → ((𝑦 ∪ {𝑧}) ∈ Omni ↔ ∀𝑔(𝑔:(𝑦 ∪ {𝑧})⟶2o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o))))
87	85, 86	ax-mp 5	. . . 4 ⊢ ((𝑦 ∪ {𝑧}) ∈ Omni ↔ ∀𝑔(𝑔:(𝑦 ∪ {𝑧})⟶2o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o)))
88	83, 87	sylibr 134	. . 3 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → (𝑦 ∪ {𝑧}) ∈ Omni)
89	88	ex 115	. 2 ⊢ (𝑦 ∈ Fin → (𝑦 ∈ Omni → (𝑦 ∪ {𝑧}) ∈ Omni))
90	1, 2, 3, 4, 11, 89	findcard2 7059	1 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ Omni)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713  ∀wal 1393   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  Vcvv 2799   ∪ cun 3195   ⊆ wss 3197  ∅c0 3491  {csn 3666  {cpr 3667   ↾ cres 4721  ⟶wf 5314  ‘cfv 5318  1_oc1o 6561  2_oc2o 6562  Fincfn 6895  Omnicomni 7309

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1o 6568  df-2o 6569  df-er 6688  df-en 6896  df-fin 6898  df-omni 7310

This theorem is referenced by:  trilpolemlt1  16439