Theorem finomni 7339

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 2294	. 2 ⊢ (𝑤 = ∅ → (𝑤 ∈ Omni ↔ ∅ ∈ Omni))
2		eleq1 2294	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ Omni ↔ 𝑦 ∈ Omni))
3		eleq1 2294	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ∈ Omni ↔ (𝑦 ∪ {𝑧}) ∈ Omni))
4		eleq1 2294	. 2 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ Omni ↔ 𝐴 ∈ Omni))
5		0ex 4216	. . . 4 ⊢ ∅ ∈ V
6		isomni 7335	. . . 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 3596	. . . . 5 ⊢ ∀𝑥 ∈ ∅ (𝑓‘𝑥) = 1o
9	8	olci 739	. . . 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 1501	. 2 ⊢ ∅ ∈ Omni
12		elun1 3374	. . . . . . . . . . . 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 5663	. . . . . . . . . . . . 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 2266	. . . . . . . . . . 11 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑔 ↾ 𝑦)‘𝑥) = ∅) → (𝑔‘𝑥) = ∅)
18		fveq2 5639	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑥 → (𝑔‘𝑢) = (𝑔‘𝑥))
19	18	eqeq1d 2240	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑥 → ((𝑔‘𝑢) = ∅ ↔ (𝑔‘𝑥) = ∅))
20	19	rspcev 2910	. . . . . . . . . . 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 740	. . . . . . . . 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 2650	. . . . . . 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 3701	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ {𝑧}
26		elun2 3375	. . . . . . . . . . . . 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 5639	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → (𝑔‘𝑢) = (𝑔‘𝑧))
30	29	eqeq1d 2240	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑧 → ((𝑔‘𝑢) = ∅ ↔ (𝑔‘𝑧) = ∅))
31	30	rspcev 2910	. . . . . . . . . . 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 740	. . . . . . . . 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 5639	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → ((𝑔 ↾ 𝑦)‘𝑥) = ((𝑔 ↾ 𝑦)‘𝑢))
37	36	eqeq1d 2240	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (((𝑔 ↾ 𝑦)‘𝑥) = 1o ↔ ((𝑔 ↾ 𝑦)‘𝑢) = 1o))
38	37	cbvralv 2767	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o ↔ ∀𝑢 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑢) = 1o)
39		fvres 5663	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑦 → ((𝑔 ↾ 𝑦)‘𝑢) = (𝑔‘𝑢))
40	39	eqeq1d 2240	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ 𝑦 → (((𝑔 ↾ 𝑦)‘𝑢) = 1o ↔ (𝑔‘𝑢) = 1o))
41	40	ralbiia 2546	. . . . . . . . . . . . 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 529	. . . . . . . . . . . 12 ⊢ (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔‘𝑧) = 1o) ∧ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o) → (𝑔‘𝑧) = 1o)
45		vex 2805	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
46	29	eqeq1d 2240	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → ((𝑔‘𝑢) = 1o ↔ (𝑔‘𝑧) = 1o))
47	45, 46	ralsn 3712	. . . . . . . . . . . 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 3389	. . . . . . . . . . 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 741	. . . . . . . . 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 5783	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (𝑔‘𝑧) ∈ 2o)
55		df2o3 6597	. . . . . . . . . 10 ⊢ 2o = {∅, 1o}
56	54, 55	eleqtrdi 2324	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (𝑔‘𝑧) ∈ {∅, 1o})
57		elpri 3692	. . . . . . . . 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 805	. . . . . . 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 2805	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
61		isomni 7335	. . . . . . . . . . . 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 2805	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
66	65	resex 5054	. . . . . . . . . 10 ⊢ (𝑔 ↾ 𝑦) ∈ V
67		feq1 5465	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → (𝑓:𝑦⟶2o ↔ (𝑔 ↾ 𝑦):𝑦⟶2o))
68		fveq1 5638	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → (𝑓‘𝑥) = ((𝑔 ↾ 𝑦)‘𝑥))
69	68	eqeq1d 2240	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → ((𝑓‘𝑥) = ∅ ↔ ((𝑔 ↾ 𝑦)‘𝑥) = ∅))
70	69	rexbidv 2533	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = ∅))
71	68	eqeq1d 2240	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → ((𝑓‘𝑥) = 1o ↔ ((𝑔 ↾ 𝑦)‘𝑥) = 1o))
72	71	ralbidv 2532	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ↾ 𝑦) → (∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝑦 ((𝑔 ↾ 𝑦)‘𝑥) = 1o))
73	70, 72	orbi12d 800	. . . . . . . . . . 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 2900	. . . . . . . . 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 3370	. . . . . . . . 9 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
78		fssres 5512	. . . . . . . . 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 725	. . . . . 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 1922	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → ∀𝑔(𝑔:(𝑦 ∪ {𝑧})⟶2o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔‘𝑢) = 1o)))
84	45	snex 4275	. . . . . 6 ⊢ {𝑧} ∈ V
85	60, 84	unex 4538	. . . . 5 ⊢ (𝑦 ∪ {𝑧}) ∈ V
86		isomni 7335	. . . . 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 7078	1 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ Omni)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715  ∀wal 1395   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  Vcvv 2802   ∪ cun 3198   ⊆ wss 3200  ∅c0 3494  {csn 3669  {cpr 3670   ↾ cres 4727  ⟶wf 5322  ‘cfv 5326  1_oc1o 6575  2_oc2o 6576  Fincfn 6909  Omnicomni 7333

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-coll 4204  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-3or 1005  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-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  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-2o 6583  df-er 6702  df-en 6910  df-fin 6912  df-omni 7334

This theorem is referenced by:  trilpolemlt1  16666