Theorem finomni 7116

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 2233	. 2 ⊢ (𝑤 = ∅ → (𝑤 ∈ Omni ↔ ∅ ∈ Omni))
2		eleq1 2233	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ Omni ↔ 𝑦 ∈ Omni))
3		eleq1 2233	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ∈ Omni ↔ (𝑦 ∪ {𝑧}) ∈ Omni))
4		eleq1 2233	. 2 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ Omni ↔ 𝐴 ∈ Omni))
5		0ex 4116	. . . 4 ⊢ ∅ ∈ V
6		isomni 7112	. . . 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 3516	. . . . 5 ⊢ ∀𝑥 ∈ ∅ (𝑓‘𝑥) = 1o
9	8	olci 727	. . . 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 1446	. 2 ⊢ ∅ ∈ Omni
12		elun1 3294	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑦 → 𝑥 ∈ (𝑦 ∪ {𝑧}))
13	12	ad2antlr 486	. . . . . . . . . . 11 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑔 ↾ 𝑦)‘𝑥) = ∅) → 𝑥 ∈ (𝑦 ∪ {𝑧}))
14		fvres 5520	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑦 → ((𝑔 ↾ 𝑦)‘𝑥) = (𝑔‘𝑥))
15	14	ad2antlr 486	. . . . . . . . . . . 12 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑔 ↾ 𝑦)‘𝑥) = ∅) → ((𝑔 ↾ 𝑦)‘𝑥) = (𝑔‘𝑥))
16		simpr 109	. . . . . . . . . . . 12 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑔 ↾ 𝑦)‘𝑥) = ∅) → ((𝑔 ↾ 𝑦)‘𝑥) = ∅)
17	15, 16	eqtr3d 2205	. . . . . . . . . . 11 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑔 ↾ 𝑦)‘𝑥) = ∅) → (𝑔‘𝑥) = ∅)
18		fveq2 5496	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑥 → (𝑔‘𝑢) = (𝑔‘𝑥))
19	18	eqeq1d 2179	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑥 → ((𝑔‘𝑢) = ∅ ↔ (𝑔‘𝑥) = ∅))
20	19	rspcev 2834	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝑦 ∪ {𝑧}) ∧ (𝑔‘𝑥) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅)
21	13, 17, 20	syl2anc 409	. . . . . . . . . 10 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑔 ↾ 𝑦)‘𝑥) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅)
22	21	orcd 728	. . . . . . . . 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 114	. . . . . . . 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 2587	. . . . . . 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 3615	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ {𝑧}
26		elun2 3295	. . . . . . . . . . . . 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 5496	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → (𝑔‘𝑢) = (𝑔‘𝑧))
30	29	eqeq1d 2179	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑧 → ((𝑔‘𝑢) = ∅ ↔ (𝑔‘𝑧) = ∅))
31	30	rspcev 2834	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝑦 ∪ {𝑧}) ∧ (𝑔‘𝑧) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅)
32	28, 31	sylan 281	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔‘𝑧) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅)
33	32	orcd 728	. . . . . . . . 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 109	. . . . . . . . . . . 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 5496	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → ((𝑔 ↾ 𝑦)‘𝑥) = ((𝑔 ↾ 𝑦)‘𝑢))
37	36	eqeq1d 2179	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (((𝑔 ↾ 𝑦)‘𝑥) = 1o ↔ ((𝑔 ↾ 𝑦)‘𝑢) = 1o))
38	37	cbvralv 2696	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o ↔ ∀𝑢 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑢) = 1o)
39		fvres 5520	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑦 → ((𝑔 ↾ 𝑦)‘𝑢) = (𝑔‘𝑢))
40	39	eqeq1d 2179	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ 𝑦 → (((𝑔 ↾ 𝑦)‘𝑢) = 1o ↔ (𝑔‘𝑢) = 1o))
41	40	ralbiia 2484	. . . . . . . . . . . . 13 ⊢ (∀𝑢 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑢) = 1o ↔ ∀𝑢 ∈ 𝑦 (𝑔‘𝑢) = 1o)
42	38, 41	bitri 183	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o ↔ ∀𝑢 ∈ 𝑦 (𝑔‘𝑢) = 1o)
43	35, 42	sylib 121	. . . . . . . . . . 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 525	. . . . . . . . . . . 12 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔‘𝑧) = 1o) ∧ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o) → (𝑔‘𝑧) = 1o)
45		vex 2733	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
46	29	eqeq1d 2179	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → ((𝑔‘𝑢) = 1o ↔ (𝑔‘𝑧) = 1o))
47	45, 46	ralsn 3626	. . . . . . . . . . . 12 ⊢ (∀𝑢 ∈ {𝑧} (𝑔‘𝑢) = 1o ↔ (𝑔‘𝑧) = 1o)
48	44, 47	sylibr 133	. . . . . . . . . . 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 3309	. . . . . . . . . . 11 ⊢ ((∀𝑢 ∈ 𝑦 (𝑔‘𝑢) = 1o ∧ ∀𝑢 ∈ {𝑧} (𝑔‘𝑢) = 1o) → ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o)
50	43, 48, 49	syl2anc 409	. . . . . . . . . 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 729	. . . . . . . . 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 114	. . . . . . . 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 109	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → 𝑔:(𝑦 ∪ {𝑧})⟶2o)
54	53, 28	ffvelrnd 5632	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (𝑔‘𝑧) ∈ 2o)
55		df2o3 6409	. . . . . . . . . 10 ⊢ 2o = {∅, 1o}
56	54, 55	eleqtrdi 2263	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (𝑔‘𝑧) ∈ {∅, 1o})
57		elpri 3606	. . . . . . . . 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 793	. . . . . . 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 2733	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
61		isomni 7112	. . . . . . . . . . . 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 119	. . . . . . . . . 10 ⊢ (𝑦 ∈ Omni → ∀𝑓(𝑓:𝑦⟶2o → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)))
64	63	adantl 275	. . . . . . . . 9 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → ∀𝑓(𝑓:𝑦⟶2o → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)))
65		vex 2733	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
66	65	resex 4932	. . . . . . . . . 10 ⊢ (𝑔 ↾ 𝑦) ∈ V
67		feq1 5330	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → (𝑓:𝑦⟶2o ↔ (𝑔 ↾ 𝑦):𝑦⟶2o))
68		fveq1 5495	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → (𝑓‘𝑥) = ((𝑔 ↾ 𝑦)‘𝑥))
69	68	eqeq1d 2179	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → ((𝑓‘𝑥) = ∅ ↔ ((𝑔 ↾ 𝑦)‘𝑥) = ∅))
70	69	rexbidv 2471	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = ∅))
71	68	eqeq1d 2179	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → ((𝑓‘𝑥) = 1o ↔ ((𝑔 ↾ 𝑦)‘𝑥) = 1o))
72	71	ralbidv 2470	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → (∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o))
73	70, 72	orbi12d 788	. . . . . . . . . . 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 233	. . . . . . . . . 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 2824	. . . . . . . . 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 3290	. . . . . . . . 9 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
78		fssres 5373	. . . . . . . . 9 ⊢ ((𝑔:(𝑦 ∪ {𝑧})⟶2o ∧ 𝑦 ⊆ (𝑦 ∪ {𝑧})) → (𝑔 ↾ 𝑦):𝑦⟶2o)
79	77, 78	mpan2 423	. . . . . . . 8 ⊢ (𝑔:(𝑦 ∪ {𝑧})⟶2o → (𝑔 ↾ 𝑦):𝑦⟶2o)
80	76, 79	impel 278	. . . . . . 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 713	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o))
82	81	ex 114	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → (𝑔:(𝑦 ∪ {𝑧})⟶2o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o)))
83	82	alrimiv 1867	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → ∀𝑔(𝑔:(𝑦 ∪ {𝑧})⟶2o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o)))
84	45	snex 4171	. . . . . 6 ⊢ {𝑧} ∈ V
85	60, 84	unex 4426	. . . . 5 ⊢ (𝑦 ∪ {𝑧}) ∈ V
86		isomni 7112	. . . . 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 133	. . 3 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → (𝑦 ∪ {𝑧}) ∈ Omni)
89	88	ex 114	. 2 ⊢ (𝑦 ∈ Fin → (𝑦 ∈ Omni → (𝑦 ∪ {𝑧}) ∈ Omni))
90	1, 2, 3, 4, 11, 89	findcard2 6867	1 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ Omni)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703  ∀wal 1346   = wceq 1348   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449  Vcvv 2730   ∪ cun 3119   ⊆ wss 3121  ∅c0 3414  {csn 3583  {cpr 3584   ↾ cres 4613  ⟶wf 5194  ‘cfv 5198  1_oc1o 6388  2_oc2o 6389  Fincfn 6718  Omnicomni 7110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1o 6395  df-2o 6396  df-er 6513  df-en 6719  df-fin 6721  df-omni 7111

This theorem is referenced by:  trilpolemlt1  14073