Theorem ralxpmap 8892

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 3476	. . 3 ⊢ 𝑔 ∈ V
2		snex 5430	. . 3 ⊢ {⟨𝐽, 𝑦⟩} ∈ V
3	1, 2	unex 7735	. 2 ⊢ (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ V
4		simpr 483	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑓 ∈ (𝑆 ↑m 𝑇))
5		elmapex 8844	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝑆 ↑m 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
6	5	adantl 480	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
7		elmapg 8835	. . . . . . . 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 231	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑓:𝑇⟶𝑆)
10		simpl 481	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝐽 ∈ 𝑇)
11	9, 10	ffvelcdmd 7086	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑓‘𝐽) ∈ 𝑆)
12		difss 4130	. . . . . . 7 ⊢ (𝑇 ∖ {𝐽}) ⊆ 𝑇
13		fssres 6756	. . . . . . 7 ⊢ ((𝑓:𝑇⟶𝑆 ∧ (𝑇 ∖ {𝐽}) ⊆ 𝑇) → (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆)
14	9, 12, 13	sylancl 584	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆)
15	5	simpld 493	. . . . . . . 8 ⊢ (𝑓 ∈ (𝑆 ↑m 𝑇) → 𝑆 ∈ V)
16	15	adantl 480	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑆 ∈ V)
17	6	simprd 494	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑇 ∈ V)
18	17	difexd 5328	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑇 ∖ {𝐽}) ∈ V)
19	16, 18	elmapd 8836	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})) ↔ (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆))
20	14, 19	mpbird 256	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))
21	9	ffnd 6717	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑓 Fn 𝑇)
22		fnsnsplit 7183	. . . . . 6 ⊢ ((𝑓 Fn 𝑇 ∧ 𝐽 ∈ 𝑇) → 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
23	21, 10, 22	syl2anc 582	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
24		opeq2 4873	. . . . . . . . 9 ⊢ (𝑦 = (𝑓‘𝐽) → ⟨𝐽, 𝑦⟩ = ⟨𝐽, (𝑓‘𝐽)⟩)
25	24	sneqd 4639	. . . . . . . 8 ⊢ (𝑦 = (𝑓‘𝐽) → {⟨𝐽, 𝑦⟩} = {⟨𝐽, (𝑓‘𝐽)⟩})
26	25	uneq2d 4162	. . . . . . 7 ⊢ (𝑦 = (𝑓‘𝐽) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}) = (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
27	26	eqeq2d 2741	. . . . . 6 ⊢ (𝑦 = (𝑓‘𝐽) → (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ↔ 𝑓 = (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩})))
28		uneq1 4155	. . . . . . 7 ⊢ (𝑔 = (𝑓 ↾ (𝑇 ∖ {𝐽})) → (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩}) = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
29	28	eqeq2d 2741	. . . . . 6 ⊢ (𝑔 = (𝑓 ↾ (𝑇 ∖ {𝐽})) → (𝑓 = (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩}) ↔ 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩})))
30	27, 29	rspc2ev 3623	. . . . 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 1369	. . . 4 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → ∃𝑦 ∈ 𝑆 ∃𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}))
32	31	ex 411	. . 3 ⊢ (𝐽 ∈ 𝑇 → (𝑓 ∈ (𝑆 ↑m 𝑇) → ∃𝑦 ∈ 𝑆 ∃𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})))
33		elmapi 8845	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})) → 𝑔:(𝑇 ∖ {𝐽})⟶𝑆)
34	33	ad2antll 725	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑔:(𝑇 ∖ {𝐽})⟶𝑆)
35		f1osng 6873	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑦 ∈ V) → {⟨𝐽, 𝑦⟩}:{𝐽}–1-1-onto→{𝑦})
36		f1of 6832	. . . . . . . . . . . 12 ⊢ ({⟨𝐽, 𝑦⟩}:{𝐽}–1-1-onto→{𝑦} → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
37	35, 36	syl 17	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑦 ∈ V) → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
38	37	elvd 3479	. . . . . . . . . 10 ⊢ (𝐽 ∈ 𝑇 → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
39	38	adantr 479	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
40		disjdifr 4471	. . . . . . . . . 10 ⊢ ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅
41	40	a1i 11	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅)
42		fun 6752	. . . . . . . . 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 834	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}))
44		simpl 481	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝐽 ∈ 𝑇)
45	44	snssd 4811	. . . . . . . . . 10 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → {𝐽} ⊆ 𝑇)
46		undifr 4481	. . . . . . . . . 10 ⊢ ({𝐽} ⊆ 𝑇 ↔ ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = 𝑇)
47	45, 46	sylib 217	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = 𝑇)
48	47	feq2d 6702	. . . . . . . 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 231	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶(𝑆 ∪ {𝑦}))
50		ssidd 4004	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑆 ⊆ 𝑆)
51		snssi 4810	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 → {𝑦} ⊆ 𝑆)
52	51	ad2antrl 724	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → {𝑦} ⊆ 𝑆)
53	50, 52	unssd 4185	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑆 ∪ {𝑦}) ⊆ 𝑆)
54	49, 53	fssd 6734	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶𝑆)
55		elmapex 8844	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})) → (𝑆 ∈ V ∧ (𝑇 ∖ {𝐽}) ∈ V))
56	55	ad2antll 725	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑆 ∈ V ∧ (𝑇 ∖ {𝐽}) ∈ V))
57	56	simpld 493	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑆 ∈ V)
58		ssun1 4171	. . . . . . . 8 ⊢ 𝑇 ⊆ (𝑇 ∪ {𝐽})
59		undif1 4474	. . . . . . . . 9 ⊢ ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = (𝑇 ∪ {𝐽})
60	56	simprd 494	. . . . . . . . . 10 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑇 ∖ {𝐽}) ∈ V)
61		snex 5430	. . . . . . . . . 10 ⊢ {𝐽} ∈ V
62		unexg 7738	. . . . . . . . . 10 ⊢ (((𝑇 ∖ {𝐽}) ∈ V ∧ {𝐽} ∈ V) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) ∈ V)
63	60, 61, 62	sylancl 584	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) ∈ V)
64	59, 63	eqeltrrid 2836	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑇 ∪ {𝐽}) ∈ V)
65		ssexg 5322	. . . . . . . 8 ⊢ ((𝑇 ⊆ (𝑇 ∪ {𝐽}) ∧ (𝑇 ∪ {𝐽}) ∈ V) → 𝑇 ∈ V)
66	58, 64, 65	sylancr 585	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑇 ∈ V)
67	57, 66	elmapd 8836	. . . . . 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 256	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆 ↑m 𝑇))
69		eleq1 2819	. . . . 5 ⊢ (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → (𝑓 ∈ (𝑆 ↑m 𝑇) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆 ↑m 𝑇)))
70	68, 69	syl5ibrcom 246	. . . 4 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → 𝑓 ∈ (𝑆 ↑m 𝑇)))
71	70	rexlimdvva 3209	. . 3 ⊢ (𝐽 ∈ 𝑇 → (∃𝑦 ∈ 𝑆 ∃𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → 𝑓 ∈ (𝑆 ↑m 𝑇)))
72	32, 71	impbid 211	. 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 480	. 2 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})) → (𝜑 ↔ 𝜓))
75	3, 72, 74	ralxpxfr2d 3633	1 ⊢ (𝐽 ∈ 𝑇 → (∀𝑓 ∈ (𝑆 ↑m 𝑇)𝜑 ↔ ∀𝑦 ∈ 𝑆 ∀𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059  ∃wrex 3068  Vcvv 3472   ∖ cdif 3944   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  {csn 4627  ⟨cop 4633   ↾ cres 5677   Fn wfn 6537  ⟶wf 6538  –1-1-onto→wf1o 6541  ‘cfv 6542  (class class class)co 7411   ↑_m cmap 8822

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824

This theorem is referenced by:  islindf4  21612