Theorem ralxpmap 8878

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 3459	. . 3 ⊢ 𝑔 ∈ V
2		snex 5397	. . 3 ⊢ {⟨𝐽, 𝑦⟩} ∈ V
3	1, 2	unex 7727	. 2 ⊢ (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ V
4		simpr 488	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑓 ∈ (𝑆 ↑m 𝑇))
5		elmapex 8829	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝑆 ↑m 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
6	5	adantl 485	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
7		elmapg 8820	. . . . . . . 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 486	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝐽 ∈ 𝑇)
11	9, 10	ffvelcdmd 7066	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑓‘𝐽) ∈ 𝑆)
12		difss 4090	. . . . . . 7 ⊢ (𝑇 ∖ {𝐽}) ⊆ 𝑇
13		fssres 6730	. . . . . . 7 ⊢ ((𝑓:𝑇⟶𝑆 ∧ (𝑇 ∖ {𝐽}) ⊆ 𝑇) → (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆)
14	9, 12, 13	sylancl 595	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆)
15	5	simpld 498	. . . . . . . 8 ⊢ (𝑓 ∈ (𝑆 ↑m 𝑇) → 𝑆 ∈ V)
16	15	adantl 485	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑆 ∈ V)
17	6	simprd 499	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑇 ∈ V)
18	17	difexd 5288	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑇 ∖ {𝐽}) ∈ V)
19	16, 18	elmapd 8821	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})) ↔ (𝑓 ↾ (𝑇 ∖ {𝐽})):(𝑇 ∖ {𝐽})⟶𝑆))
20	14, 19	mpbird 259	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → (𝑓 ↾ (𝑇 ∖ {𝐽})) ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))
21	9	ffnd 6692	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑓 Fn 𝑇)
22		fnsnsplit 7168	. . . . . 6 ⊢ ((𝑓 Fn 𝑇 ∧ 𝐽 ∈ 𝑇) → 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
23	21, 10, 22	syl2anc 593	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
24		opeq2 4833	. . . . . . . . 9 ⊢ (𝑦 = (𝑓‘𝐽) → ⟨𝐽, 𝑦⟩ = ⟨𝐽, (𝑓‘𝐽)⟩)
25	24	sneqd 4595	. . . . . . . 8 ⊢ (𝑦 = (𝑓‘𝐽) → {⟨𝐽, 𝑦⟩} = {⟨𝐽, (𝑓‘𝐽)⟩})
26	25	uneq2d 4122	. . . . . . 7 ⊢ (𝑦 = (𝑓‘𝐽) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}) = (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
27	26	eqeq2d 2774	. . . . . 6 ⊢ (𝑦 = (𝑓‘𝐽) → (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ↔ 𝑓 = (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩})))
28		uneq1 4115	. . . . . . 7 ⊢ (𝑔 = (𝑓 ↾ (𝑇 ∖ {𝐽})) → (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩}) = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩}))
29	28	eqeq2d 2774	. . . . . 6 ⊢ (𝑔 = (𝑓 ↾ (𝑇 ∖ {𝐽})) → (𝑓 = (𝑔 ∪ {⟨𝐽, (𝑓‘𝐽)⟩}) ↔ 𝑓 = ((𝑓 ↾ (𝑇 ∖ {𝐽})) ∪ {⟨𝐽, (𝑓‘𝐽)⟩})))
30	27, 29	rspc2ev 3595	. . . . 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 1392	. . . 4 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 ∈ (𝑆 ↑m 𝑇)) → ∃𝑦 ∈ 𝑆 ∃𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}))
32	31	ex 416	. . 3 ⊢ (𝐽 ∈ 𝑇 → (𝑓 ∈ (𝑆 ↑m 𝑇) → ∃𝑦 ∈ 𝑆 ∃𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})))
33		elmapi 8830	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})) → 𝑔:(𝑇 ∖ {𝐽})⟶𝑆)
34	33	ad2antll 739	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑔:(𝑇 ∖ {𝐽})⟶𝑆)
35		f1osng 6849	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑦 ∈ V) → {⟨𝐽, 𝑦⟩}:{𝐽}–1-1-onto→{𝑦})
36		f1of 6806	. . . . . . . . . . . 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 3461	. . . . . . . . . 10 ⊢ (𝐽 ∈ 𝑇 → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
39	38	adantr 484	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → {⟨𝐽, 𝑦⟩}:{𝐽}⟶{𝑦})
40		disjdifr 4428	. . . . . . . . . 10 ⊢ ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅
41	40	a1i 11	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∩ {𝐽}) = ∅)
42		fun 6726	. . . . . . . . 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 848	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):((𝑇 ∖ {𝐽}) ∪ {𝐽})⟶(𝑆 ∪ {𝑦}))
44		simpl 486	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝐽 ∈ 𝑇)
45	44	snssd 4746	. . . . . . . . . 10 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → {𝐽} ⊆ 𝑇)
46		undifr 4438	. . . . . . . . . 10 ⊢ ({𝐽} ⊆ 𝑇 ↔ ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = 𝑇)
47	45, 46	sylib 220	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = 𝑇)
48	47	feq2d 6675	. . . . . . . 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 234	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶(𝑆 ∪ {𝑦}))
50		ssidd 3960	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑆 ⊆ 𝑆)
51		snssi 4745	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 → {𝑦} ⊆ 𝑆)
52	51	ad2antrl 738	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → {𝑦} ⊆ 𝑆)
53	50, 52	unssd 4145	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑆 ∪ {𝑦}) ⊆ 𝑆)
54	49, 53	fssd 6709	. . . . . 6 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}):𝑇⟶𝑆)
55		elmapex 8829	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})) → (𝑆 ∈ V ∧ (𝑇 ∖ {𝐽}) ∈ V))
56	55	ad2antll 739	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑆 ∈ V ∧ (𝑇 ∖ {𝐽}) ∈ V))
57	56	simpld 498	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑆 ∈ V)
58		ssun1 4131	. . . . . . . 8 ⊢ 𝑇 ⊆ (𝑇 ∪ {𝐽})
59		undif1 4431	. . . . . . . . 9 ⊢ ((𝑇 ∖ {𝐽}) ∪ {𝐽}) = (𝑇 ∪ {𝐽})
60	56	simprd 499	. . . . . . . . . 10 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑇 ∖ {𝐽}) ∈ V)
61		snex 5397	. . . . . . . . . 10 ⊢ {𝐽} ∈ V
62		unexg 7726	. . . . . . . . . 10 ⊢ (((𝑇 ∖ {𝐽}) ∈ V ∧ {𝐽} ∈ V) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) ∈ V)
63	60, 61, 62	sylancl 595	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → ((𝑇 ∖ {𝐽}) ∪ {𝐽}) ∈ V)
64	59, 63	eqeltrrid 2868	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑇 ∪ {𝐽}) ∈ V)
65		ssexg 5280	. . . . . . . 8 ⊢ ((𝑇 ⊆ (𝑇 ∪ {𝐽}) ∧ (𝑇 ∪ {𝐽}) ∈ V) → 𝑇 ∈ V)
66	58, 64, 65	sylancr 596	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → 𝑇 ∈ V)
67	57, 66	elmapd 8821	. . . . . 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 259	. . . . 5 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆 ↑m 𝑇))
69		eleq1 2851	. . . . 5 ⊢ (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → (𝑓 ∈ (𝑆 ↑m 𝑇) ↔ (𝑔 ∪ {⟨𝐽, 𝑦⟩}) ∈ (𝑆 ↑m 𝑇)))
70	68, 69	syl5ibrcom 249	. . . 4 ⊢ ((𝐽 ∈ 𝑇 ∧ (𝑦 ∈ 𝑆 ∧ 𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽})))) → (𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → 𝑓 ∈ (𝑆 ↑m 𝑇)))
71	70	rexlimdvva 3220	. . 3 ⊢ (𝐽 ∈ 𝑇 → (∃𝑦 ∈ 𝑆 ∃𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩}) → 𝑓 ∈ (𝑆 ↑m 𝑇)))
72	32, 71	impbid 214	. 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 485	. 2 ⊢ ((𝐽 ∈ 𝑇 ∧ 𝑓 = (𝑔 ∪ {⟨𝐽, 𝑦⟩})) → (𝜑 ↔ 𝜓))
75	3, 72, 74	ralxpxfr2d 3606	1 ⊢ (𝐽 ∈ 𝑇 → (∀𝑓 ∈ (𝑆 ↑m 𝑇)𝜑 ↔ ∀𝑦 ∈ 𝑆 ∀𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1561   ∈ wcel 2143  ∀wral 3077  ∃wrex 3087  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  {csn 4583  ⟨cop 4589   ↾ cres 5650   Fn wfn 6516  ⟶wf 6517  –1-1-onto→wf1o 6520  ‘cfv 6521  (class class class)co 7396   ↑_m cmap 8808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-map 8810

This theorem is referenced by:  islindf4  21897