Theorem brapply 34910

Description: Binary relation form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

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 5432	. . . 4 ⊢ {(𝐴 “ {𝐵})} ∈ V
2	1	inex1 5318	. . 3 ⊢ ({(𝐴 “ {𝐵})} ∩ Singletons ) ∈ V
3		unieq 4920	. . . . 5 ⊢ (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) → ∪ 𝑥 = ∪ ({(𝐴 “ {𝐵})} ∩ Singletons ))
4	3	unieqd 4923	. . . 4 ⊢ (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) → ∪ ∪ 𝑥 = ∪ ∪ ({(𝐴 “ {𝐵})} ∩ Singletons ))
5	4	eqeq2d 2744	. . 3 ⊢ (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) → (𝐶 = ∪ ∪ 𝑥 ↔ 𝐶 = ∪ ∪ ({(𝐴 “ {𝐵})} ∩ Singletons )))
6	2, 5	ceqsexv 3526	. 2 ⊢ (∃𝑥(𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) ∧ 𝐶 = ∪ ∪ 𝑥) ↔ 𝐶 = ∪ ∪ ({(𝐴 “ {𝐵})} ∩ Singletons ))
7		df-apply 34845	. . . 4 ⊢ Apply = (( Bigcup ∘ Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))
8	7	breqi 5155	. . 3 ⊢ (⟨𝐴, 𝐵⟩Apply𝐶 ↔ ⟨𝐴, 𝐵⟩(( Bigcup ∘ Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))𝐶)
9		opex 5465	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
10		brapply.3	. . . 4 ⊢ 𝐶 ∈ V
11	9, 10	brco 5871	. . 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 3479	. . . . . . 7 ⊢ 𝑥 ∈ V
13	9, 12	brco 5871	. . . . . 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 3479	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
15	9, 14	brco 5871	. . . . . . . . 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 3479	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
19	16, 17, 18	brpprod3a 34858	. . . . . . . . . . . 12 ⊢ (⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧 ↔ ∃𝑎∃𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐵Singleton𝑏))
20		3anrot 1101	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐵Singleton𝑏) ↔ (𝐴 I 𝑎 ∧ 𝐵Singleton𝑏 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
21		vex 3479	. . . . . . . . . . . . . . . . 17 ⊢ 𝑎 ∈ V
22	21	ideq 5853	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 I 𝑎 ↔ 𝐴 = 𝑎)
23		eqcom 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = 𝑎 ↔ 𝑎 = 𝐴)
24	22, 23	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (𝐴 I 𝑎 ↔ 𝑎 = 𝐴)
25		vex 3479	. . . . . . . . . . . . . . . 16 ⊢ 𝑏 ∈ V
26	17, 25	brsingle 34889	. . . . . . . . . . . . . . 15 ⊢ (𝐵Singleton𝑏 ↔ 𝑏 = {𝐵})
27		biid 261	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑎, 𝑏⟩)
28	24, 26, 27	3anbi123i 1156	. . . . . . . . . . . . . 14 ⊢ ((𝐴 I 𝑎 ∧ 𝐵Singleton𝑏 ∧ 𝑧 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
29	20, 28	bitri 275	. . . . . . . . . . . . 13 ⊢ ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐵Singleton𝑏) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
30	29	2exbii 1852	. . . . . . . . . . . 12 ⊢ (∃𝑎∃𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐵Singleton𝑏) ↔ ∃𝑎∃𝑏(𝑎 = 𝐴 ∧ 𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
31		snex 5432	. . . . . . . . . . . . 13 ⊢ {𝐵} ∈ V
32		opeq1 4874	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
33	32	eqeq2d 2744	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝐴, 𝑏⟩))
34		opeq2 4875	. . . . . . . . . . . . . 14 ⊢ (𝑏 = {𝐵} → ⟨𝐴, 𝑏⟩ = ⟨𝐴, {𝐵}⟩)
35	34	eqeq2d 2744	. . . . . . . . . . . . 13 ⊢ (𝑏 = {𝐵} → (𝑧 = ⟨𝐴, 𝑏⟩ ↔ 𝑧 = ⟨𝐴, {𝐵}⟩))
36	16, 31, 33, 35	ceqsex2v 3531	. . . . . . . . . . . 12 ⊢ (∃𝑎∃𝑏(𝑎 = 𝐴 ∧ 𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩) ↔ 𝑧 = ⟨𝐴, {𝐵}⟩)
37	19, 30, 36	3bitri 297	. . . . . . . . . . 11 ⊢ (⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧 ↔ 𝑧 = ⟨𝐴, {𝐵}⟩)
38	37	anbi1i 625	. . . . . . . . . 10 ⊢ ((⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧 ∧ 𝑧(Singleton ∘ Img)𝑦) ↔ (𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦))
39	38	exbii 1851	. . . . . . . . 9 ⊢ (∃𝑧(⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧 ∧ 𝑧(Singleton ∘ Img)𝑦) ↔ ∃𝑧(𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦))
40		opex 5465	. . . . . . . . . . 11 ⊢ ⟨𝐴, {𝐵}⟩ ∈ V
41		breq1 5152	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝐴, {𝐵}⟩ → (𝑧(Singleton ∘ Img)𝑦 ↔ ⟨𝐴, {𝐵}⟩(Singleton ∘ Img)𝑦))
42	40, 41	ceqsexv 3526	. . . . . . . . . 10 ⊢ (∃𝑧(𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦) ↔ ⟨𝐴, {𝐵}⟩(Singleton ∘ Img)𝑦)
43	40, 14	brco 5871	. . . . . . . . . 10 ⊢ (⟨𝐴, {𝐵}⟩(Singleton ∘ Img)𝑦 ↔ ∃𝑥(⟨𝐴, {𝐵}⟩Img𝑥 ∧ 𝑥Singleton𝑦))
44	16, 31, 12	brimg 34909	. . . . . . . . . . . . 13 ⊢ (⟨𝐴, {𝐵}⟩Img𝑥 ↔ 𝑥 = (𝐴 “ {𝐵}))
45	12, 14	brsingle 34889	. . . . . . . . . . . . 13 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥})
46	44, 45	anbi12i 628	. . . . . . . . . . . 12 ⊢ ((⟨𝐴, {𝐵}⟩Img𝑥 ∧ 𝑥Singleton𝑦) ↔ (𝑥 = (𝐴 “ {𝐵}) ∧ 𝑦 = {𝑥}))
47	46	exbii 1851	. . . . . . . . . . 11 ⊢ (∃𝑥(⟨𝐴, {𝐵}⟩Img𝑥 ∧ 𝑥Singleton𝑦) ↔ ∃𝑥(𝑥 = (𝐴 “ {𝐵}) ∧ 𝑦 = {𝑥}))
48	16	imaex 7907	. . . . . . . . . . . 12 ⊢ (𝐴 “ {𝐵}) ∈ V
49		sneq 4639	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐴 “ {𝐵}) → {𝑥} = {(𝐴 “ {𝐵})})
50	49	eqeq2d 2744	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐴 “ {𝐵}) → (𝑦 = {𝑥} ↔ 𝑦 = {(𝐴 “ {𝐵})}))
51	48, 50	ceqsexv 3526	. . . . . . . . . . 11 ⊢ (∃𝑥(𝑥 = (𝐴 “ {𝐵}) ∧ 𝑦 = {𝑥}) ↔ 𝑦 = {(𝐴 “ {𝐵})})
52	47, 51	bitri 275	. . . . . . . . . 10 ⊢ (∃𝑥(⟨𝐴, {𝐵}⟩Img𝑥 ∧ 𝑥Singleton𝑦) ↔ 𝑦 = {(𝐴 “ {𝐵})})
53	42, 43, 52	3bitri 297	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦) ↔ 𝑦 = {(𝐴 “ {𝐵})})
54	15, 39, 53	3bitri 297	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩((Singleton ∘ Img) ∘ pprod( I , Singleton))𝑦 ↔ 𝑦 = {(𝐴 “ {𝐵})})
55		eqid 2733	. . . . . . . . 9 ⊢ ((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))
56		brxp 5726	. . . . . . . . . 10 ⊢ (𝑦(V × V)𝑥 ↔ (𝑦 ∈ V ∧ 𝑥 ∈ V))
57	14, 12, 56	mpbir2an 710	. . . . . . . . 9 ⊢ 𝑦(V × V)𝑥
58		epel 5584	. . . . . . . . . . 11 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦)
59	58	anbi1ci 627	. . . . . . . . . 10 ⊢ ((𝑧 ∈ Singletons ∧ 𝑧 E 𝑦) ↔ (𝑧 ∈ 𝑦 ∧ 𝑧 ∈ Singletons ))
60	14	brresi 5991	. . . . . . . . . 10 ⊢ (𝑧( E ↾ Singletons )𝑦 ↔ (𝑧 ∈ Singletons ∧ 𝑧 E 𝑦))
61		elin 3965	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑦 ∩ Singletons ) ↔ (𝑧 ∈ 𝑦 ∧ 𝑧 ∈ Singletons ))
62	59, 60, 61	3bitr4ri 304	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑦 ∩ Singletons ) ↔ 𝑧( E ↾ Singletons )𝑦)
63	14, 12, 55, 57, 62	brtxpsd3 34868	. . . . . . . 8 ⊢ (𝑦((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))𝑥 ↔ 𝑥 = (𝑦 ∩ Singletons ))
64	54, 63	anbi12i 628	. . . . . . 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 )))
65	64	exbii 1851	. . . . . 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 )))
66		ineq1 4206	. . . . . . . 8 ⊢ (𝑦 = {(𝐴 “ {𝐵})} → (𝑦 ∩ Singletons ) = ({(𝐴 “ {𝐵})} ∩ Singletons ))
67	66	eqeq2d 2744	. . . . . . 7 ⊢ (𝑦 = {(𝐴 “ {𝐵})} → (𝑥 = (𝑦 ∩ Singletons ) ↔ 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons )))
68	1, 67	ceqsexv 3526	. . . . . 6 ⊢ (∃𝑦(𝑦 = {(𝐴 “ {𝐵})} ∧ 𝑥 = (𝑦 ∩ Singletons )) ↔ 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
69	13, 65, 68	3bitri 297	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩(((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton)))𝑥 ↔ 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
70	12, 10	brco 5871	. . . . . 6 ⊢ (𝑥( Bigcup ∘ Bigcup )𝐶 ↔ ∃𝑦(𝑥 Bigcup 𝑦 ∧ 𝑦 Bigcup 𝐶))
71	14	brbigcup 34870	. . . . . . . . 9 ⊢ (𝑥 Bigcup 𝑦 ↔ ∪ 𝑥 = 𝑦)
72		eqcom 2740	. . . . . . . . 9 ⊢ (∪ 𝑥 = 𝑦 ↔ 𝑦 = ∪ 𝑥)
73	71, 72	bitri 275	. . . . . . . 8 ⊢ (𝑥 Bigcup 𝑦 ↔ 𝑦 = ∪ 𝑥)
74	10	brbigcup 34870	. . . . . . . . 9 ⊢ (𝑦 Bigcup 𝐶 ↔ ∪ 𝑦 = 𝐶)
75		eqcom 2740	. . . . . . . . 9 ⊢ (∪ 𝑦 = 𝐶 ↔ 𝐶 = ∪ 𝑦)
76	74, 75	bitri 275	. . . . . . . 8 ⊢ (𝑦 Bigcup 𝐶 ↔ 𝐶 = ∪ 𝑦)
77	73, 76	anbi12i 628	. . . . . . 7 ⊢ ((𝑥 Bigcup 𝑦 ∧ 𝑦 Bigcup 𝐶) ↔ (𝑦 = ∪ 𝑥 ∧ 𝐶 = ∪ 𝑦))
78	77	exbii 1851	. . . . . 6 ⊢ (∃𝑦(𝑥 Bigcup 𝑦 ∧ 𝑦 Bigcup 𝐶) ↔ ∃𝑦(𝑦 = ∪ 𝑥 ∧ 𝐶 = ∪ 𝑦))
79		vuniex 7729	. . . . . . 7 ⊢ ∪ 𝑥 ∈ V
80		unieq 4920	. . . . . . . 8 ⊢ (𝑦 = ∪ 𝑥 → ∪ 𝑦 = ∪ ∪ 𝑥)
81	80	eqeq2d 2744	. . . . . . 7 ⊢ (𝑦 = ∪ 𝑥 → (𝐶 = ∪ 𝑦 ↔ 𝐶 = ∪ ∪ 𝑥))
82	79, 81	ceqsexv 3526	. . . . . 6 ⊢ (∃𝑦(𝑦 = ∪ 𝑥 ∧ 𝐶 = ∪ 𝑦) ↔ 𝐶 = ∪ ∪ 𝑥)
83	70, 78, 82	3bitri 297	. . . . 5 ⊢ (𝑥( Bigcup ∘ Bigcup )𝐶 ↔ 𝐶 = ∪ ∪ 𝑥)
84	69, 83	anbi12i 628	. . . 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))
85	84	exbii 1851	. . 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))
86	8, 11, 85	3bitri 297	. 2 ⊢ (⟨𝐴, 𝐵⟩Apply𝐶 ↔ ∃𝑥(𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) ∧ 𝐶 = ∪ ∪ 𝑥))
87		dffv5 34896	. . 3 ⊢ (𝐴‘𝐵) = ∪ ∪ ({(𝐴 “ {𝐵})} ∩ Singletons )
88	87	eqeq2i 2746	. 2 ⊢ (𝐶 = (𝐴‘𝐵) ↔ 𝐶 = ∪ ∪ ({(𝐴 “ {𝐵})} ∩ Singletons ))
89	6, 86, 88	3bitr4i 303	1 ⊢ (⟨𝐴, 𝐵⟩Apply𝐶 ↔ 𝐶 = (𝐴‘𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107  Vcvv 3475   ∖ cdif 3946   ∩ cin 3948   △ csymdif 4242  {csn 4629  ⟨cop 4635  ∪ cuni 4909   class class class wbr 5149   I cid 5574   E cep 5580   × cxp 5675  ran crn 5678   ↾ cres 5679   “ cima 5680   ∘ ccom 5681  ‘cfv 6544   ⊗ ctxp 34802  pprodcpprod 34803   Bigcup cbigcup 34806  Singletoncsingle 34810   Singletons csingles 34811  Imgcimg 34814  Applycapply 34817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-symdif 4243  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-eprel 5581  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-1st 7975  df-2nd 7976  df-txp 34826  df-pprod 34827  df-bigcup 34830  df-singleton 34834  df-singles 34835  df-image 34836  df-cart 34837  df-img 34838  df-apply 34845

This theorem is referenced by:  dfrecs2  34922  dfrdg4  34923