Theorem brimg 34239

Description: Binary relation form of the Img 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
brimg.1	⊢ 𝐴 ∈ V
brimg.2	⊢ 𝐵 ∈ V
brimg.3	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
brimg	⊢ (⟨𝐴, 𝐵⟩Img𝐶 ↔ 𝐶 = (𝐴 “ 𝐵))

Proof of Theorem brimg

Dummy variables 𝑎 𝑏 𝑝 𝑞 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-img 34168	. . 3 ⊢ Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)
2	1	breqi 5080	. 2 ⊢ (⟨𝐴, 𝐵⟩Img𝐶 ↔ ⟨𝐴, 𝐵⟩(Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)𝐶)
3		opex 5379	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
4		brimg.3	. . . 4 ⊢ 𝐶 ∈ V
5	3, 4	brco 5779	. . 3 ⊢ (⟨𝐴, 𝐵⟩(Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)𝐶 ↔ ∃𝑎(⟨𝐴, 𝐵⟩Cart𝑎 ∧ 𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶))
6		brimg.1	. . . . . 6 ⊢ 𝐴 ∈ V
7		brimg.2	. . . . . 6 ⊢ 𝐵 ∈ V
8		vex 3436	. . . . . 6 ⊢ 𝑎 ∈ V
9	6, 7, 8	brcart 34234	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩Cart𝑎 ↔ 𝑎 = (𝐴 × 𝐵))
10	9	anbi1i 624	. . . 4 ⊢ ((⟨𝐴, 𝐵⟩Cart𝑎 ∧ 𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶) ↔ (𝑎 = (𝐴 × 𝐵) ∧ 𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶))
11	10	exbii 1850	. . 3 ⊢ (∃𝑎(⟨𝐴, 𝐵⟩Cart𝑎 ∧ 𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶) ↔ ∃𝑎(𝑎 = (𝐴 × 𝐵) ∧ 𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶))
12	6, 7	xpex 7603	. . . 4 ⊢ (𝐴 × 𝐵) ∈ V
13		breq1 5077	. . . 4 ⊢ (𝑎 = (𝐴 × 𝐵) → (𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶 ↔ (𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶))
14	12, 13	ceqsexv 3479	. . 3 ⊢ (∃𝑎(𝑎 = (𝐴 × 𝐵) ∧ 𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶) ↔ (𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶)
15	5, 11, 14	3bitri 297	. 2 ⊢ (⟨𝐴, 𝐵⟩(Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)𝐶 ↔ (𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶)
16	12, 4	brimage 34228	. . 3 ⊢ ((𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶 ↔ 𝐶 = (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)))
17		19.42v 1957	. . . . . . . 8 ⊢ (∃𝑎(𝑏 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)) ↔ (𝑏 ∈ 𝐵 ∧ ∃𝑎(𝑎 ∈ 𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
18		anass 469	. . . . . . . . . . 11 ⊢ (((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
19		an21 641	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (𝑏 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
20	19	anbi2i 623	. . . . . . . . . . 11 ⊢ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)) ↔ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))))
21	18, 20	bitri 274	. . . . . . . . . 10 ⊢ (((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))))
22	21	2exbii 1851	. . . . . . . . 9 ⊢ (∃𝑝∃𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑝∃𝑎(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))))
23		excom 2162	. . . . . . . . 9 ⊢ (∃𝑝∃𝑎(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))) ↔ ∃𝑎∃𝑝(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))))
24		opex 5379	. . . . . . . . . . 11 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
25		breq1 5077	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥 ↔ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
26	25	anbi2d 629	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝑎 ∈ 𝐴 ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (𝑎 ∈ 𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
27	26	anbi2d 629	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝑏 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)) ↔ (𝑏 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))))
28	24, 27	ceqsexv 3479	. . . . . . . . . 10 ⊢ (∃𝑝(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))) ↔ (𝑏 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
29	28	exbii 1850	. . . . . . . . 9 ⊢ (∃𝑎∃𝑝(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))) ↔ ∃𝑎(𝑏 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
30	22, 23, 29	3bitri 297	. . . . . . . 8 ⊢ (∃𝑝∃𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑎(𝑏 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
31		df-br 5075	. . . . . . . . . 10 ⊢ (𝑏𝐴𝑥 ↔ ⟨𝑏, 𝑥⟩ ∈ 𝐴)
32		risset 3194	. . . . . . . . . . 11 ⊢ (⟨𝑏, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑎 ∈ 𝐴 𝑎 = ⟨𝑏, 𝑥⟩)
33		vex 3436	. . . . . . . . . . . . . . . 16 ⊢ 𝑏 ∈ V
34	33	brresi 5900	. . . . . . . . . . . . . . 15 ⊢ (𝑎(1st ↾ (V × V))𝑏 ↔ (𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏))
35		df-br 5075	. . . . . . . . . . . . . . 15 ⊢ (𝑎(1st ↾ (V × V))𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ (1st ↾ (V × V)))
36	34, 35	bitr3i 276	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏) ↔ ⟨𝑎, 𝑏⟩ ∈ (1st ↾ (V × V)))
37	36	anbi1i 624	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏) ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ (1st ↾ (V × V)) ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥))
38		elvv 5661	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ (V × V) ↔ ∃𝑝∃𝑞 𝑎 = ⟨𝑝, 𝑞⟩)
39	38	anbi1i 624	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ (V × V) ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ (∃𝑝∃𝑞 𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)))
40		anass 469	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏) ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (𝑎 ∈ (V × V) ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)))
41		ancom 461	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑝 = 𝑏 ∧ 𝑞 = 𝑥)) ↔ ((𝑝 = 𝑏 ∧ 𝑞 = 𝑥) ∧ 𝑎 = ⟨𝑝, 𝑞⟩))
42		breq1 5077	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = ⟨𝑝, 𝑞⟩ → (𝑎1st 𝑏 ↔ ⟨𝑝, 𝑞⟩1st 𝑏))
43		opeq1 4804	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = ⟨𝑝, 𝑞⟩ → ⟨𝑎, 𝑏⟩ = ⟨⟨𝑝, 𝑞⟩, 𝑏⟩)
44	43	breq1d 5084	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = ⟨𝑝, 𝑞⟩ → (⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥 ↔ ⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥))
45	42, 44	anbi12d 631	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = ⟨𝑝, 𝑞⟩ → ((𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (⟨𝑝, 𝑞⟩1st 𝑏 ∧ ⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥)))
46		vex 3436	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑝 ∈ V
47		vex 3436	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑞 ∈ V
48	46, 47	br1steq 33745	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨𝑝, 𝑞⟩1st 𝑏 ↔ 𝑏 = 𝑝)
49		equcom 2021	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = 𝑝 ↔ 𝑝 = 𝑏)
50	48, 49	bitri 274	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨𝑝, 𝑞⟩1st 𝑏 ↔ 𝑝 = 𝑏)
51		opex 5379	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨⟨𝑝, 𝑞⟩, 𝑏⟩ ∈ V
52		vex 3436	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑥 ∈ V
53	51, 52	brco 5779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥 ↔ ∃𝑎(⟨⟨𝑝, 𝑞⟩, 𝑏⟩1st 𝑎 ∧ 𝑎2nd 𝑥))
54		opex 5379	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ⟨𝑝, 𝑞⟩ ∈ V
55	54, 33	br1steq 33745	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨⟨𝑝, 𝑞⟩, 𝑏⟩1st 𝑎 ↔ 𝑎 = ⟨𝑝, 𝑞⟩)
56	55	anbi1i 624	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((⟨⟨𝑝, 𝑞⟩, 𝑏⟩1st 𝑎 ∧ 𝑎2nd 𝑥) ↔ (𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥))
57	56	exbii 1850	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑎(⟨⟨𝑝, 𝑞⟩, 𝑏⟩1st 𝑎 ∧ 𝑎2nd 𝑥) ↔ ∃𝑎(𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥))
58	53, 57	bitri 274	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥 ↔ ∃𝑎(𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥))
59		breq1 5077	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = ⟨𝑝, 𝑞⟩ → (𝑎2nd 𝑥 ↔ ⟨𝑝, 𝑞⟩2nd 𝑥))
60	54, 59	ceqsexv 3479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑎(𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥) ↔ ⟨𝑝, 𝑞⟩2nd 𝑥)
61	46, 47	br2ndeq 33746	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑝, 𝑞⟩2nd 𝑥 ↔ 𝑥 = 𝑞)
62	60, 61	bitri 274	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑎(𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥) ↔ 𝑥 = 𝑞)
63		equcom 2021	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑞 ↔ 𝑞 = 𝑥)
64	58, 62, 63	3bitri 297	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥 ↔ 𝑞 = 𝑥)
65	50, 64	anbi12i 627	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨𝑝, 𝑞⟩1st 𝑏 ∧ ⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (𝑝 = 𝑏 ∧ 𝑞 = 𝑥))
66	45, 65	bitrdi 287	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = ⟨𝑝, 𝑞⟩ → ((𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (𝑝 = 𝑏 ∧ 𝑞 = 𝑥)))
67	66	pm5.32i 575	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ (𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑝 = 𝑏 ∧ 𝑞 = 𝑥)))
68		df-3an 1088	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑝 = 𝑏 ∧ 𝑞 = 𝑥 ∧ 𝑎 = ⟨𝑝, 𝑞⟩) ↔ ((𝑝 = 𝑏 ∧ 𝑞 = 𝑥) ∧ 𝑎 = ⟨𝑝, 𝑞⟩))
69	41, 67, 68	3bitr4i 303	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ (𝑝 = 𝑏 ∧ 𝑞 = 𝑥 ∧ 𝑎 = ⟨𝑝, 𝑞⟩))
70	69	2exbii 1851	. . . . . . . . . . . . . . 15 ⊢ (∃𝑝∃𝑞(𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ ∃𝑝∃𝑞(𝑝 = 𝑏 ∧ 𝑞 = 𝑥 ∧ 𝑎 = ⟨𝑝, 𝑞⟩))
71		19.41vv 1954	. . . . . . . . . . . . . . 15 ⊢ (∃𝑝∃𝑞(𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ (∃𝑝∃𝑞 𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)))
72		opeq1 4804	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = 𝑏 → ⟨𝑝, 𝑞⟩ = ⟨𝑏, 𝑞⟩)
73	72	eqeq2d 2749	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = 𝑏 → (𝑎 = ⟨𝑝, 𝑞⟩ ↔ 𝑎 = ⟨𝑏, 𝑞⟩))
74		opeq2 4805	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 = 𝑥 → ⟨𝑏, 𝑞⟩ = ⟨𝑏, 𝑥⟩)
75	74	eqeq2d 2749	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 𝑥 → (𝑎 = ⟨𝑏, 𝑞⟩ ↔ 𝑎 = ⟨𝑏, 𝑥⟩))
76	33, 52, 73, 75	ceqsex2v 3483	. . . . . . . . . . . . . . 15 ⊢ (∃𝑝∃𝑞(𝑝 = 𝑏 ∧ 𝑞 = 𝑥 ∧ 𝑎 = ⟨𝑝, 𝑞⟩) ↔ 𝑎 = ⟨𝑏, 𝑥⟩)
77	70, 71, 76	3bitr3ri 302	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ⟨𝑏, 𝑥⟩ ↔ (∃𝑝∃𝑞 𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)))
78	39, 40, 77	3bitr4ri 304	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑏, 𝑥⟩ ↔ ((𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏) ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥))
79	52	brresi 5900	. . . . . . . . . . . . 13 ⊢ (⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥 ↔ (⟨𝑎, 𝑏⟩ ∈ (1st ↾ (V × V)) ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥))
80	37, 78, 79	3bitr4i 303	. . . . . . . . . . . 12 ⊢ (𝑎 = ⟨𝑏, 𝑥⟩ ↔ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)
81	80	rexbii 3181	. . . . . . . . . . 11 ⊢ (∃𝑎 ∈ 𝐴 𝑎 = ⟨𝑏, 𝑥⟩ ↔ ∃𝑎 ∈ 𝐴 ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)
82	32, 81	bitri 274	. . . . . . . . . 10 ⊢ (⟨𝑏, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑎 ∈ 𝐴 ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)
83		df-rex 3070	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ 𝐴 ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥 ↔ ∃𝑎(𝑎 ∈ 𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
84	31, 82, 83	3bitri 297	. . . . . . . . 9 ⊢ (𝑏𝐴𝑥 ↔ ∃𝑎(𝑎 ∈ 𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
85	84	anbi2i 623	. . . . . . . 8 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑏𝐴𝑥) ↔ (𝑏 ∈ 𝐵 ∧ ∃𝑎(𝑎 ∈ 𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
86	17, 30, 85	3bitr4ri 304	. . . . . . 7 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑏𝐴𝑥) ↔ ∃𝑝∃𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
87	86	exbii 1850	. . . . . 6 ⊢ (∃𝑏(𝑏 ∈ 𝐵 ∧ 𝑏𝐴𝑥) ↔ ∃𝑏∃𝑝∃𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
88	52	elima2 5975	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 “ 𝐵) ↔ ∃𝑏(𝑏 ∈ 𝐵 ∧ 𝑏𝐴𝑥))
89	52	elima2 5975	. . . . . . 7 ⊢ (𝑥 ∈ (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) ↔ ∃𝑝(𝑝 ∈ (𝐴 × 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
90		elxp 5612	. . . . . . . . . 10 ⊢ (𝑝 ∈ (𝐴 × 𝐵) ↔ ∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)))
91	90	anbi1i 624	. . . . . . . . 9 ⊢ ((𝑝 ∈ (𝐴 × 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
92		19.41vv 1954	. . . . . . . . 9 ⊢ (∃𝑎∃𝑏((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
93	91, 92	bitr4i 277	. . . . . . . 8 ⊢ ((𝑝 ∈ (𝐴 × 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑎∃𝑏((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
94	93	exbii 1850	. . . . . . 7 ⊢ (∃𝑝(𝑝 ∈ (𝐴 × 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑝∃𝑎∃𝑏((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
95		exrot3 2165	. . . . . . . 8 ⊢ (∃𝑝∃𝑎∃𝑏((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑎∃𝑏∃𝑝((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
96		exrot3 2165	. . . . . . . 8 ⊢ (∃𝑎∃𝑏∃𝑝((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑏∃𝑝∃𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
97	95, 96	bitri 274	. . . . . . 7 ⊢ (∃𝑝∃𝑎∃𝑏((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑏∃𝑝∃𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
98	89, 94, 97	3bitri 297	. . . . . 6 ⊢ (𝑥 ∈ (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) ↔ ∃𝑏∃𝑝∃𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
99	87, 88, 98	3bitr4ri 304	. . . . 5 ⊢ (𝑥 ∈ (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) ↔ 𝑥 ∈ (𝐴 “ 𝐵))
100	99	eqriv 2735	. . . 4 ⊢ (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) = (𝐴 “ 𝐵)
101	100	eqeq2i 2751	. . 3 ⊢ (𝐶 = (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) ↔ 𝐶 = (𝐴 “ 𝐵))
102	16, 101	bitri 274	. 2 ⊢ ((𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶 ↔ 𝐶 = (𝐴 “ 𝐵))
103	2, 15, 102	3bitri 297	1 ⊢ (⟨𝐴, 𝐵⟩Img𝐶 ↔ 𝐶 = (𝐴 “ 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432  ⟨cop 4567   class class class wbr 5074   × cxp 5587   ↾ cres 5591   “ cima 5592   ∘ ccom 5593  1^st c1st 7829  2^nd c2nd 7830  Imagecimage 34142  Cartccart 34143  Imgcimg 34144

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-symdif 4176  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-eprel 5495  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-1st 7831  df-2nd 7832  df-txp 34156  df-pprod 34157  df-image 34166  df-cart 34167  df-img 34168

This theorem is referenced by:  brapply  34240  dfrdg4  34253