Theorem brapply 32045

Description: Binary relation form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

Ref	Expression
brapply.1	⊢ 𝐴 ∈ V
brapply.2	⊢ 𝐵 ∈ V
brapply.3	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
brapply	⊢ (⟨𝐴, 𝐵⟩Apply𝐶 ↔ 𝐶 = (𝐴‘𝐵))

Proof of Theorem brapply

Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4908	. . . 4 ⊢ {(𝐴 “ {𝐵})} ∈ V
2	1	inex1 4799	. . 3 ⊢ ({(𝐴 “ {𝐵})} ∩ Singletons ) ∈ V
3		unieq 4444	. . . . 5 ⊢ (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) → ∪ 𝑥 = ∪ ({(𝐴 “ {𝐵})} ∩ Singletons ))
4	3	unieqd 4446	. . . 4 ⊢ (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) → ∪ ∪ 𝑥 = ∪ ∪ ({(𝐴 “ {𝐵})} ∩ Singletons ))
5	4	eqeq2d 2632	. . 3 ⊢ (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) → (𝐶 = ∪ ∪ 𝑥 ↔ 𝐶 = ∪ ∪ ({(𝐴 “ {𝐵})} ∩ Singletons )))
6	2, 5	ceqsexv 3242	. 2 ⊢ (∃𝑥(𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) ∧ 𝐶 = ∪ ∪ 𝑥) ↔ 𝐶 = ∪ ∪ ({(𝐴 “ {𝐵})} ∩ Singletons ))
7		df-apply 31980	. . . 4 ⊢ Apply = (( Bigcup ∘ Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))
8	7	breqi 4659	. . 3 ⊢ (⟨𝐴, 𝐵⟩Apply𝐶 ↔ ⟨𝐴, 𝐵⟩(( Bigcup ∘ Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))𝐶)
9		opex 4932	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
10		brapply.3	. . . 4 ⊢ 𝐶 ∈ V
11	9, 10	brco 5292	. . 3 ⊢ (⟨𝐴, 𝐵⟩(( Bigcup ∘ Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))𝐶 ↔ ∃𝑥(⟨𝐴, 𝐵⟩(((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton)))𝑥 ∧ 𝑥( Bigcup ∘ Bigcup )𝐶))
12		vex 3203	. . . . . . 7 ⊢ 𝑥 ∈ V
13	9, 12	brco 5292	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩(((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton)))𝑥 ↔ ∃𝑦(⟨𝐴, 𝐵⟩((Singleton ∘ Img) ∘ pprod( I , Singleton))𝑦 ∧ 𝑦((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))𝑥))
14		vex 3203	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
15	9, 14	brco 5292	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩((Singleton ∘ Img) ∘ pprod( I , Singleton))𝑦 ↔ ∃𝑧(⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧 ∧ 𝑧(Singleton ∘ Img)𝑦))
16		brapply.1	. . . . . . . . . . . . 13 ⊢ 𝐴 ∈ V
17		brapply.2	. . . . . . . . . . . . 13 ⊢ 𝐵 ∈ V
18		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
19	16, 17, 18	brpprod3a 31993	. . . . . . . . . . . 12 ⊢ (⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧 ↔ ∃𝑎∃𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐵Singleton𝑏))
20		3anrot 1043	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐵Singleton𝑏) ↔ (𝐴 I 𝑎 ∧ 𝐵Singleton𝑏 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
21		vex 3203	. . . . . . . . . . . . . . . . 17 ⊢ 𝑎 ∈ V
22	21	ideq 5274	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 I 𝑎 ↔ 𝐴 = 𝑎)
23		eqcom 2629	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = 𝑎 ↔ 𝑎 = 𝐴)
24	22, 23	bitri 264	. . . . . . . . . . . . . . 15 ⊢ (𝐴 I 𝑎 ↔ 𝑎 = 𝐴)
25		vex 3203	. . . . . . . . . . . . . . . 16 ⊢ 𝑏 ∈ V
26	17, 25	brsingle 32024	. . . . . . . . . . . . . . 15 ⊢ (𝐵Singleton𝑏 ↔ 𝑏 = {𝐵})
27		biid 251	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑎, 𝑏⟩)
28	24, 26, 27	3anbi123i 1251	. . . . . . . . . . . . . 14 ⊢ ((𝐴 I 𝑎 ∧ 𝐵Singleton𝑏 ∧ 𝑧 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
29	20, 28	bitri 264	. . . . . . . . . . . . 13 ⊢ ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐵Singleton𝑏) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
30	29	2exbii 1775	. . . . . . . . . . . 12 ⊢ (∃𝑎∃𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐵Singleton𝑏) ↔ ∃𝑎∃𝑏(𝑎 = 𝐴 ∧ 𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
31		snex 4908	. . . . . . . . . . . . 13 ⊢ {𝐵} ∈ V
32		opeq1 4402	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
33	32	eqeq2d 2632	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝐴, 𝑏⟩))
34		opeq2 4403	. . . . . . . . . . . . . 14 ⊢ (𝑏 = {𝐵} → ⟨𝐴, 𝑏⟩ = ⟨𝐴, {𝐵}⟩)
35	34	eqeq2d 2632	. . . . . . . . . . . . 13 ⊢ (𝑏 = {𝐵} → (𝑧 = ⟨𝐴, 𝑏⟩ ↔ 𝑧 = ⟨𝐴, {𝐵}⟩))
36	16, 31, 33, 35	ceqsex2v 3245	. . . . . . . . . . . 12 ⊢ (∃𝑎∃𝑏(𝑎 = 𝐴 ∧ 𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩) ↔ 𝑧 = ⟨𝐴, {𝐵}⟩)
37	19, 30, 36	3bitri 286	. . . . . . . . . . 11 ⊢ (⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧 ↔ 𝑧 = ⟨𝐴, {𝐵}⟩)
38	37	anbi1i 731	. . . . . . . . . 10 ⊢ ((⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧 ∧ 𝑧(Singleton ∘ Img)𝑦) ↔ (𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦))
39	38	exbii 1774	. . . . . . . . 9 ⊢ (∃𝑧(⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧 ∧ 𝑧(Singleton ∘ Img)𝑦) ↔ ∃𝑧(𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦))
40		opex 4932	. . . . . . . . . . 11 ⊢ ⟨𝐴, {𝐵}⟩ ∈ V
41		breq1 4656	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝐴, {𝐵}⟩ → (𝑧(Singleton ∘ Img)𝑦 ↔ ⟨𝐴, {𝐵}⟩(Singleton ∘ Img)𝑦))
42	40, 41	ceqsexv 3242	. . . . . . . . . 10 ⊢ (∃𝑧(𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦) ↔ ⟨𝐴, {𝐵}⟩(Singleton ∘ Img)𝑦)
43	40, 14	brco 5292	. . . . . . . . . 10 ⊢ (⟨𝐴, {𝐵}⟩(Singleton ∘ Img)𝑦 ↔ ∃𝑥(⟨𝐴, {𝐵}⟩Img𝑥 ∧ 𝑥Singleton𝑦))
44	16, 31, 12	brimg 32044	. . . . . . . . . . . . 13 ⊢ (⟨𝐴, {𝐵}⟩Img𝑥 ↔ 𝑥 = (𝐴 “ {𝐵}))
45	12, 14	brsingle 32024	. . . . . . . . . . . . 13 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥})
46	44, 45	anbi12i 733	. . . . . . . . . . . 12 ⊢ ((⟨𝐴, {𝐵}⟩Img𝑥 ∧ 𝑥Singleton𝑦) ↔ (𝑥 = (𝐴 “ {𝐵}) ∧ 𝑦 = {𝑥}))
47	46	exbii 1774	. . . . . . . . . . 11 ⊢ (∃𝑥(⟨𝐴, {𝐵}⟩Img𝑥 ∧ 𝑥Singleton𝑦) ↔ ∃𝑥(𝑥 = (𝐴 “ {𝐵}) ∧ 𝑦 = {𝑥}))
48		imaexg 7103	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ V → (𝐴 “ {𝐵}) ∈ V)
49	16, 48	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝐴 “ {𝐵}) ∈ V
50		sneq 4187	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐴 “ {𝐵}) → {𝑥} = {(𝐴 “ {𝐵})})
51	50	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐴 “ {𝐵}) → (𝑦 = {𝑥} ↔ 𝑦 = {(𝐴 “ {𝐵})}))
52	49, 51	ceqsexv 3242	. . . . . . . . . . 11 ⊢ (∃𝑥(𝑥 = (𝐴 “ {𝐵}) ∧ 𝑦 = {𝑥}) ↔ 𝑦 = {(𝐴 “ {𝐵})})
53	47, 52	bitri 264	. . . . . . . . . 10 ⊢ (∃𝑥(⟨𝐴, {𝐵}⟩Img𝑥 ∧ 𝑥Singleton𝑦) ↔ 𝑦 = {(𝐴 “ {𝐵})})
54	42, 43, 53	3bitri 286	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦) ↔ 𝑦 = {(𝐴 “ {𝐵})})
55	15, 39, 54	3bitri 286	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩((Singleton ∘ Img) ∘ pprod( I , Singleton))𝑦 ↔ 𝑦 = {(𝐴 “ {𝐵})})
56		eqid 2622	. . . . . . . . 9 ⊢ ((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))
57		brxp 5147	. . . . . . . . . 10 ⊢ (𝑦(V × V)𝑥 ↔ (𝑦 ∈ V ∧ 𝑥 ∈ V))
58	14, 12, 57	mpbir2an 955	. . . . . . . . 9 ⊢ 𝑦(V × V)𝑥
59		epel 5032	. . . . . . . . . . 11 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦)
60	59	anbi1i 731	. . . . . . . . . 10 ⊢ ((𝑧 E 𝑦 ∧ 𝑧 ∈ Singletons ) ↔ (𝑧 ∈ 𝑦 ∧ 𝑧 ∈ Singletons ))
61	14	brres 5402	. . . . . . . . . 10 ⊢ (𝑧( E ↾ Singletons )𝑦 ↔ (𝑧 E 𝑦 ∧ 𝑧 ∈ Singletons ))
62		elin 3796	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑦 ∩ Singletons ) ↔ (𝑧 ∈ 𝑦 ∧ 𝑧 ∈ Singletons ))
63	60, 61, 62	3bitr4ri 293	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑦 ∩ Singletons ) ↔ 𝑧( E ↾ Singletons )𝑦)
64	14, 12, 56, 58, 63	brtxpsd3 32003	. . . . . . . 8 ⊢ (𝑦((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))𝑥 ↔ 𝑥 = (𝑦 ∩ Singletons ))
65	55, 64	anbi12i 733	. . . . . . 7 ⊢ ((⟨𝐴, 𝐵⟩((Singleton ∘ Img) ∘ pprod( I , Singleton))𝑦 ∧ 𝑦((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))𝑥) ↔ (𝑦 = {(𝐴 “ {𝐵})} ∧ 𝑥 = (𝑦 ∩ Singletons )))
66	65	exbii 1774	. . . . . 6 ⊢ (∃𝑦(⟨𝐴, 𝐵⟩((Singleton ∘ Img) ∘ pprod( I , Singleton))𝑦 ∧ 𝑦((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))𝑥) ↔ ∃𝑦(𝑦 = {(𝐴 “ {𝐵})} ∧ 𝑥 = (𝑦 ∩ Singletons )))
67		ineq1 3807	. . . . . . . 8 ⊢ (𝑦 = {(𝐴 “ {𝐵})} → (𝑦 ∩ Singletons ) = ({(𝐴 “ {𝐵})} ∩ Singletons ))
68	67	eqeq2d 2632	. . . . . . 7 ⊢ (𝑦 = {(𝐴 “ {𝐵})} → (𝑥 = (𝑦 ∩ Singletons ) ↔ 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons )))
69	1, 68	ceqsexv 3242	. . . . . 6 ⊢ (∃𝑦(𝑦 = {(𝐴 “ {𝐵})} ∧ 𝑥 = (𝑦 ∩ Singletons )) ↔ 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
70	13, 66, 69	3bitri 286	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩(((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton)))𝑥 ↔ 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
71	12, 10	brco 5292	. . . . . 6 ⊢ (𝑥( Bigcup ∘ Bigcup )𝐶 ↔ ∃𝑦(𝑥 Bigcup 𝑦 ∧ 𝑦 Bigcup 𝐶))
72	14	brbigcup 32005	. . . . . . . . 9 ⊢ (𝑥 Bigcup 𝑦 ↔ ∪ 𝑥 = 𝑦)
73		eqcom 2629	. . . . . . . . 9 ⊢ (∪ 𝑥 = 𝑦 ↔ 𝑦 = ∪ 𝑥)
74	72, 73	bitri 264	. . . . . . . 8 ⊢ (𝑥 Bigcup 𝑦 ↔ 𝑦 = ∪ 𝑥)
75	10	brbigcup 32005	. . . . . . . . 9 ⊢ (𝑦 Bigcup 𝐶 ↔ ∪ 𝑦 = 𝐶)
76		eqcom 2629	. . . . . . . . 9 ⊢ (∪ 𝑦 = 𝐶 ↔ 𝐶 = ∪ 𝑦)
77	75, 76	bitri 264	. . . . . . . 8 ⊢ (𝑦 Bigcup 𝐶 ↔ 𝐶 = ∪ 𝑦)
78	74, 77	anbi12i 733	. . . . . . 7 ⊢ ((𝑥 Bigcup 𝑦 ∧ 𝑦 Bigcup 𝐶) ↔ (𝑦 = ∪ 𝑥 ∧ 𝐶 = ∪ 𝑦))
79	78	exbii 1774	. . . . . 6 ⊢ (∃𝑦(𝑥 Bigcup 𝑦 ∧ 𝑦 Bigcup 𝐶) ↔ ∃𝑦(𝑦 = ∪ 𝑥 ∧ 𝐶 = ∪ 𝑦))
80		vuniex 6954	. . . . . . 7 ⊢ ∪ 𝑥 ∈ V
81		unieq 4444	. . . . . . . 8 ⊢ (𝑦 = ∪ 𝑥 → ∪ 𝑦 = ∪ ∪ 𝑥)
82	81	eqeq2d 2632	. . . . . . 7 ⊢ (𝑦 = ∪ 𝑥 → (𝐶 = ∪ 𝑦 ↔ 𝐶 = ∪ ∪ 𝑥))
83	80, 82	ceqsexv 3242	. . . . . 6 ⊢ (∃𝑦(𝑦 = ∪ 𝑥 ∧ 𝐶 = ∪ 𝑦) ↔ 𝐶 = ∪ ∪ 𝑥)
84	71, 79, 83	3bitri 286	. . . . 5 ⊢ (𝑥( Bigcup ∘ Bigcup )𝐶 ↔ 𝐶 = ∪ ∪ 𝑥)
85	70, 84	anbi12i 733	. . . 4 ⊢ ((⟨𝐴, 𝐵⟩(((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton)))𝑥 ∧ 𝑥( Bigcup ∘ Bigcup )𝐶) ↔ (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) ∧ 𝐶 = ∪ ∪ 𝑥))
86	85	exbii 1774	. . 3 ⊢ (∃𝑥(⟨𝐴, 𝐵⟩(((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton)))𝑥 ∧ 𝑥( Bigcup ∘ Bigcup )𝐶) ↔ ∃𝑥(𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) ∧ 𝐶 = ∪ ∪ 𝑥))
87	8, 11, 86	3bitri 286	. 2 ⊢ (⟨𝐴, 𝐵⟩Apply𝐶 ↔ ∃𝑥(𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) ∧ 𝐶 = ∪ ∪ 𝑥))
88		dffv5 32031	. . 3 ⊢ (𝐴‘𝐵) = ∪ ∪ ({(𝐴 “ {𝐵})} ∩ Singletons )
89	88	eqeq2i 2634	. 2 ⊢ (𝐶 = (𝐴‘𝐵) ↔ 𝐶 = ∪ ∪ ({(𝐴 “ {𝐵})} ∩ Singletons ))
90	6, 87, 89	3bitr4i 292	1 ⊢ (⟨𝐴, 𝐵⟩Apply𝐶 ↔ 𝐶 = (𝐴‘𝐵))

Syntax hints:   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   △ csymdif 3843  {csn 4177  ⟨cop 4183  ∪ cuni 4436   class class class wbr 4653   I cid 5023   E cep 5028   × cxp 5112  ran crn 5115   ↾ cres 5116   “ cima 5117   ∘ ccom 5118  ‘cfv 5888   ⊗ ctxp 31937  pprodcpprod 31938   Bigcup cbigcup 31941  Singletoncsingle 31945   Singletons csingles 31946  Imgcimg 31949  Applycapply 31952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-symdif 3844  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-pprod 31962  df-bigcup 31965  df-singleton 31969  df-singles 31970  df-image 31971  df-cart 31972  df-img 31973  df-apply 31980

This theorem is referenced by:  dfrecs2  32057  dfrdg4  32058