Theorem ralxpmap 8906

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 3457	. . 3 ⊢ 𝑔 ∈ V
2		snex 5408	. . 3 ⊢ {⟨𝐽, 𝑦⟩} ∈ V
3	1, 2	unex 7749	. 2 ⊢ (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ V
4		simpr 490	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑓 ∈ (𝑆 ↑m 𝑇))
5		elmapex 8850	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝑆 ↑m 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
6	5	adantl 487	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
7		elmapg 8841	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑓 ∈ (𝑆 ↑m 𝑇) ↔ 𝑓:𝑇⟶𝑆))
8	6, 7	syl 18	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑓 ∈ (𝑆 ↑m 𝑇) ↔ 𝑓:𝑇⟶𝑆))
9	4, 8	mpbid 235	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑓:𝑇⟶𝑆)
10		simpl 488	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝐽 ∈ 𝑇)
11	9, 10	ffvelcdmd 7081	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑓‘𝐽) ∈ 𝑆)
12		difss 4086	. . . . . . 7 ⊢ (𝑇 ∖ {𝐽}) ⊆ 𝑇
13		fssres 6745	. . . . . . 7 ⊢ ((𝑓:𝑇⟶𝑆 ∧ (𝑇 ∖ {𝐽}) ⊆ 𝑇) → (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆)
14	9, 12, 13	sylancl 598	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆)
15	5	simpld 500	. . . . . . . 8 ⊢ (𝑓 ∈ (𝑆 ↑m 𝑇) → 𝑆 ∈ V)
16	15	adantl 487	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑆 ∈ V)
17	6	simprd 501	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑇 ∈ V)
18	17	difexd 5300	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑇 ∖ {𝐽}) ∈ V)
19	16, 18	elmapd 8842	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})) ↔ (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆))
20	14, 19	mpbird 260	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))
21	9	ffnd 6707	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑓 Fn 𝑇)
22		fnsnsplit 7185	. . . . . 6 ⊢ ((𝑓 Fn 𝑇 ∧ 𝐽 ∈ 𝑇) → 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
23	21, 10, 22	syl2anc 596	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
24		opeq2 4837	. . . . . . . . 9 ⊢ (𝑦 = (𝑓‘𝐽) → ⟨𝐽, 𝑦⟩ = ⟨𝐽, (𝑓‘𝐽)⟩)
25	24	sneqd 4599	. . . . . . . 8 ⊢ (𝑦 = (𝑓‘𝐽) → {⟨𝐽, 𝑦⟩} = {⟨𝐽, (𝑓‘𝐽)⟩})
26	25	uneq2d 4118	. . . . . . 7 ⊢ (𝑦 = (𝑓‘𝐽) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}) = (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
27	26	eqeq2d 2773	. . . . . 6 ⊢ (𝑦 = (𝑓‘𝐽) → (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ↔ 𝑓 = (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩})))
28		uneq1 4111	. . . . . . 7 ⊢ (𝑔 = (𝑓 ↾ (𝑇 ∖ {𝐽})) → (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩}) = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
29	28	eqeq2d 2773	. . . . . 6 ⊢ (𝑔 = (𝑓 ↾ (𝑇 ∖ {𝐽})) → (𝑓 = (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩}) ↔ 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩})))
30	27, 29	rspc2ev 3592	. . . . 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⟩}))
31	11, 20, 23, 30	syl3anc 1398	. . . 4 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → ∃𝑦 ∈ 𝑆 ∃𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}))
32	31	ex 418	. . 3 ⊢ (𝐽 ∈ 𝑇 → (𝑓 ∈ (𝑆 ↑m 𝑇) → ∃𝑦 ∈ 𝑆 ∃𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})))
33		elmapi 8851	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})) → 𝑔:(𝑇 ∖ {𝐽})⟶𝑆)
34	33	ad2antll 742	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑔:(𝑇 ∖ {𝐽})⟶𝑆)
35		f1osng 6864	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑦 ∈ V) → {⟨𝐽, 𝑦⟩}:{𝐽}–1-1-onto→{𝑦})
36		f1of 6821	. . . . . . . . . . . 12 ⊢ ({⟨𝐽, 𝑦⟩}:{𝐽}–1-1-onto→{𝑦} → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
37	35, 36	syl 18	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑦 ∈ V) → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
38	37	elvd 3459	. . . . . . . . . 10 ⊢ (𝐽 ∈ 𝑇 → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
39	38	adantr 486	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
40		disjdifr 4430	. . . . . . . . . 10 ⊢ ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅
41	40	a1i 11	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅)
42		fun 6741	. . . . . . . . 9 ⊢ (((𝑔:(𝑇 ∖ {𝐽})⟶𝑆 ∧ {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦}) ∧ ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}))
43	34, 39, 41, 42	syl21anc 851	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}))
44		simpl 488	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝐽 ∈ 𝑇)
45	44	snssd 4750	. . . . . . . . . 10 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → {𝐽} ⊆ 𝑇)
46		undifr 4442	. . . . . . . . . 10 ⊢ ({𝐽} ⊆ 𝑇 ↔ ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = 𝑇)
47	45, 46	sylib 221	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = 𝑇)
48	47	feq2d 6690	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶(𝑆 ∪ {𝑦})))
49	43, 48	mpbid 235	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶(𝑆 ∪ {𝑦}))
50		ssidd 3957	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑆 ⊆ 𝑆)
51		snssi 4749	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 → {𝑦} ⊆ 𝑆)
52	51	ad2antrl 741	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → {𝑦} ⊆ 𝑆)
53	50, 52	unssd 4141	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑆 ∪ {𝑦}) ⊆ 𝑆)
54	49, 53	fssd 6724	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶𝑆)
55		elmapex 8850	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})) → (𝑆 ∈ V ∧ (𝑇 ∖ {𝐽}) ∈ V))
56	55	ad2antll 742	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑆 ∈ V ∧ (𝑇 ∖ {𝐽}) ∈ V))
57	56	simpld 500	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑆 ∈ V)
58		ssun1 4127	. . . . . . . 8 ⊢ 𝑇 ⊆ (𝑇 ∪ {𝐽})
59		undif1 4433	. . . . . . . . 9 ⊢ ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = (𝑇 ∪ {𝐽})
60	56	simprd 501	. . . . . . . . . 10 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑇 ∖ {𝐽}) ∈ V)
61		snex 5408	. . . . . . . . . 10 ⊢ {𝐽} ∈ V
62		unexg 7748	. . . . . . . . . 10 ⊢ (((𝑇 ∖ {𝐽}) ∈ V ∧ {𝐽} ∈ V) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) ∈ V)
63	60, 61, 62	sylancl 598	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) ∈ V)
64	59, 63	eqeltrrid 2867	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑇 ∪ {𝐽}) ∈ V)
65		ssexg 5288	. . . . . . . 8 ⊢ ((𝑇 ⊆ (𝑇 ∪ {𝐽}) ∧ (𝑇 ∪ {𝐽}) ∈ V) → 𝑇 ∈ V)
66	58, 64, 65	sylancr 599	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑇 ∈ V)
67	57, 66	elmapd 8842	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆 ↑m 𝑇) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶𝑆))
68	54, 67	mpbird 260	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆 ↑m 𝑇))
69		eleq1 2850	. . . . 5 ⊢ (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → (𝑓 ∈ (𝑆 ↑m 𝑇) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆 ↑m 𝑇)))
70	68, 69	syl5ibrcom 250	. . . 4 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → 𝑓 ∈ (𝑆 ↑m 𝑇)))
71	70	rexlimdvva 3221	. . 3 ⊢ (𝐽 ∈ 𝑇 → (∃𝑦 ∈ 𝑆 ∃𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → 𝑓 ∈ (𝑆 ↑m 𝑇)))
72	32, 71	impbid 215	. 2 ⊢ (𝐽 ∈ 𝑇 → (𝑓 ∈ (𝑆 ↑m 𝑇) ↔ ∃𝑦 ∈ 𝑆 ∃𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})))
73		ralxpmap.j	. . 3 ⊢ (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → (𝜑 ↔ 𝜓))
74	73	adantl 487	. 2 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})) → (𝜑 ↔ 𝜓))
75	3, 72, 74	ralxpxfr2d 3603	1 ⊢ (𝐽 ∈ 𝑇 → (∀𝑓 ∈ (𝑆 ↑m 𝑇)𝜑 ↔ ∀𝑦 ∈ 𝑆 ∀𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  {csn 4587  ⟨cop 4593   ↾ cres 5661   Fn wfn 6532  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7416   ↑_m cmap 8829

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-map 8831

This theorem is used by:  islindf4  22052