Theorem ralxpmap 8463

Description: Quantification over functions in terms of quantification over values and punctured functions. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)

Hypothesis

Ref	Expression
ralxpmap.j	⊢ (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ralxpmap	⊢ (𝐽 ∈ 𝑇 → (∀𝑓 ∈ (𝑆 ↑m 𝑇)𝜑 ↔ ∀𝑦 ∈ 𝑆 ∀𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝜓))

Distinct variable groups:   𝜑,𝑔,𝑦   𝜓,𝑓   𝑓,𝐽,𝑔,𝑦   𝑆,𝑓,𝑔,𝑦   𝑇,𝑓,𝑔,𝑦

Allowed substitution hints:   𝜑(𝑓)   𝜓(𝑦,𝑔)

Proof of Theorem ralxpmap

Step	Hyp	Ref	Expression
1		vex 3500	. . 3 ⊢ 𝑔 ∈ V
2		snex 5335	. . 3 ⊢ {⟨𝐽, 𝑦⟩} ∈ V
3	1, 2	unex 7472	. 2 ⊢ (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ V
4		simpr 487	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑓 ∈ (𝑆 ↑m 𝑇))
5		elmapex 8430	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝑆 ↑m 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
6	5	adantl 484	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
7		elmapg 8422	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑓 ∈ (𝑆 ↑m 𝑇) ↔ 𝑓:𝑇⟶𝑆))
8	6, 7	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑓 ∈ (𝑆 ↑m 𝑇) ↔ 𝑓:𝑇⟶𝑆))
9	4, 8	mpbid 234	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑓:𝑇⟶𝑆)
10		simpl 485	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝐽 ∈ 𝑇)
11	9, 10	ffvelrnd 6855	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑓‘𝐽) ∈ 𝑆)
12		difss 4111	. . . . . . 7 ⊢ (𝑇 ∖ {𝐽}) ⊆ 𝑇
13		fssres 6547	. . . . . . 7 ⊢ ((𝑓:𝑇⟶𝑆 ∧ (𝑇 ∖ {𝐽}) ⊆ 𝑇) → (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆)
14	9, 12, 13	sylancl 588	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆)
15	5	simpld 497	. . . . . . . 8 ⊢ (𝑓 ∈ (𝑆 ↑m 𝑇) → 𝑆 ∈ V)
16	15	adantl 484	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑆 ∈ V)
17	6	simprd 498	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑇 ∈ V)
18		difexg 5234	. . . . . . . 8 ⊢ (𝑇 ∈ V → (𝑇 ∖ {𝐽}) ∈ V)
19	17, 18	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑇 ∖ {𝐽}) ∈ V)
20	16, 19	elmapd 8423	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})) ↔ (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆))
21	14, 20	mpbird 259	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))
22	9	ffnd 6518	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑓 Fn 𝑇)
23		fnsnsplit 6949	. . . . . 6 ⊢ ((𝑓 Fn 𝑇 ∧ 𝐽 ∈ 𝑇) → 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
24	22, 10, 23	syl2anc 586	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
25		opeq2 4807	. . . . . . . . 9 ⊢ (𝑦 = (𝑓‘𝐽) → ⟨𝐽, 𝑦⟩ = ⟨𝐽, (𝑓‘𝐽)⟩)
26	25	sneqd 4582	. . . . . . . 8 ⊢ (𝑦 = (𝑓‘𝐽) → {⟨𝐽, 𝑦⟩} = {⟨𝐽, (𝑓‘𝐽)⟩})
27	26	uneq2d 4142	. . . . . . 7 ⊢ (𝑦 = (𝑓‘𝐽) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}) = (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
28	27	eqeq2d 2835	. . . . . 6 ⊢ (𝑦 = (𝑓‘𝐽) → (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ↔ 𝑓 = (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩})))
29		uneq1 4135	. . . . . . 7 ⊢ (𝑔 = (𝑓 ↾ (𝑇 ∖ {𝐽})) → (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩}) = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
30	29	eqeq2d 2835	. . . . . 6 ⊢ (𝑔 = (𝑓 ↾ (𝑇 ∖ {𝐽})) → (𝑓 = (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩}) ↔ 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩})))
31	28, 30	rspc2ev 3638	. . . . 5 ⊢ (((𝑓‘𝐽) ∈ 𝑆 ∧ (𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})) ∧ 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩})) → ∃𝑦 ∈ 𝑆 ∃𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}))
32	11, 21, 24, 31	syl3anc 1367	. . . 4 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → ∃𝑦 ∈ 𝑆 ∃𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}))
33	32	ex 415	. . 3 ⊢ (𝐽 ∈ 𝑇 → (𝑓 ∈ (𝑆 ↑m 𝑇) → ∃𝑦 ∈ 𝑆 ∃𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})))
34		elmapi 8431	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})) → 𝑔:(𝑇 ∖ {𝐽})⟶𝑆)
35	34	ad2antll 727	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑔:(𝑇 ∖ {𝐽})⟶𝑆)
36		f1osng 6658	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑦 ∈ V) → {⟨𝐽, 𝑦⟩}:{𝐽}–1-1-onto→{𝑦})
37		f1of 6618	. . . . . . . . . . . 12 ⊢ ({⟨𝐽, 𝑦⟩}:{𝐽}–1-1-onto→{𝑦} → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
38	36, 37	syl 17	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑦 ∈ V) → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
39	38	elvd 3503	. . . . . . . . . 10 ⊢ (𝐽 ∈ 𝑇 → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
40	39	adantr 483	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
41		incom 4181	. . . . . . . . . . 11 ⊢ ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ({𝐽} ∩ (𝑇 ∖ {𝐽}))
42		disjdif 4424	. . . . . . . . . . 11 ⊢ ({𝐽} ∩ (𝑇 ∖ {𝐽})) = ∅
43	41, 42	eqtri 2847	. . . . . . . . . 10 ⊢ ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅
44	43	a1i 11	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅)
45		fun 6543	. . . . . . . . 9 ⊢ (((𝑔:(𝑇 ∖ {𝐽})⟶𝑆 ∧ {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦}) ∧ ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}))
46	35, 40, 44, 45	syl21anc 835	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}))
47		uncom 4132	. . . . . . . . . 10 ⊢ ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = ({𝐽} ∪ (𝑇 ∖ {𝐽}))
48		simpl 485	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝐽 ∈ 𝑇)
49	48	snssd 4745	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → {𝐽} ⊆ 𝑇)
50		undif 4433	. . . . . . . . . . 11 ⊢ ({𝐽} ⊆ 𝑇 ↔ ({𝐽} ∪ (𝑇 ∖ {𝐽})) = 𝑇)
51	49, 50	sylib 220	. . . . . . . . . 10 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ({𝐽} ∪ (𝑇 ∖ {𝐽})) = 𝑇)
52	47, 51	syl5eq 2871	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = 𝑇)
53	52	feq2d 6503	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶(𝑆 ∪ {𝑦})))
54	46, 53	mpbid 234	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶(𝑆 ∪ {𝑦}))
55		ssidd 3993	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑆 ⊆ 𝑆)
56		snssi 4744	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 → {𝑦} ⊆ 𝑆)
57	56	ad2antrl 726	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → {𝑦} ⊆ 𝑆)
58	55, 57	unssd 4165	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑆 ∪ {𝑦}) ⊆ 𝑆)
59	54, 58	fssd 6531	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶𝑆)
60		elmapex 8430	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})) → (𝑆 ∈ V ∧ (𝑇 ∖ {𝐽}) ∈ V))
61	60	ad2antll 727	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑆 ∈ V ∧ (𝑇 ∖ {𝐽}) ∈ V))
62	61	simpld 497	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑆 ∈ V)
63		ssun1 4151	. . . . . . . 8 ⊢ 𝑇 ⊆ (𝑇 ∪ {𝐽})
64		undif1 4427	. . . . . . . . 9 ⊢ ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = (𝑇 ∪ {𝐽})
65	61	simprd 498	. . . . . . . . . 10 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑇 ∖ {𝐽}) ∈ V)
66		snex 5335	. . . . . . . . . 10 ⊢ {𝐽} ∈ V
67		unexg 7475	. . . . . . . . . 10 ⊢ (((𝑇 ∖ {𝐽}) ∈ V ∧ {𝐽} ∈ V) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) ∈ V)
68	65, 66, 67	sylancl 588	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) ∈ V)
69	64, 68	eqeltrrid 2921	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑇 ∪ {𝐽}) ∈ V)
70		ssexg 5230	. . . . . . . 8 ⊢ ((𝑇 ⊆ (𝑇 ∪ {𝐽}) ∧ (𝑇 ∪ {𝐽}) ∈ V) → 𝑇 ∈ V)
71	63, 69, 70	sylancr 589	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑇 ∈ V)
72	62, 71	elmapd 8423	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆 ↑m 𝑇) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶𝑆))
73	59, 72	mpbird 259	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆 ↑m 𝑇))
74		eleq1 2903	. . . . 5 ⊢ (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → (𝑓 ∈ (𝑆 ↑m 𝑇) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆 ↑m 𝑇)))
75	73, 74	syl5ibrcom 249	. . . 4 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → 𝑓 ∈ (𝑆 ↑m 𝑇)))
76	75	rexlimdvva 3297	. . 3 ⊢ (𝐽 ∈ 𝑇 → (∃𝑦 ∈ 𝑆 ∃𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → 𝑓 ∈ (𝑆 ↑m 𝑇)))
77	33, 76	impbid 214	. 2 ⊢ (𝐽 ∈ 𝑇 → (𝑓 ∈ (𝑆 ↑m 𝑇) ↔ ∃𝑦 ∈ 𝑆 ∃𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})))
78		ralxpmap.j	. . 3 ⊢ (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → (𝜑 ↔ 𝜓))
79	78	adantl 484	. 2 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})) → (𝜑 ↔ 𝜓))
80	3, 77, 79	ralxpxfr2d 3642	1 ⊢ (𝐽 ∈ 𝑇 → (∀𝑓 ∈ (𝑆 ↑m 𝑇)𝜑 ↔ ∀𝑦 ∈ 𝑆 ∀𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3141  ∃wrex 3142  Vcvv 3497   ∖ cdif 3936   ∪ cun 3937   ∩ cin 3938   ⊆ wss 3939  ∅c0 4294  {csn 4570  ⟨cop 4576   ↾ cres 5560   Fn wfn 6353  ⟶wf 6354  –1-1-onto→wf1o 6357  ‘cfv 6358  (class class class)co 7159   ↑_m cmap 8409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-map 8411

This theorem is referenced by:  islindf4  20985