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Theorem brimg 32169
Description: Binary relation form of the Img function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brimg.1 𝐴 ∈ V
brimg.2 𝐵 ∈ V
brimg.3 𝐶 ∈ V
Assertion
Ref Expression
brimg (⟨𝐴, 𝐵⟩Img𝐶𝐶 = (𝐴𝐵))

Proof of Theorem brimg
Dummy variables 𝑎 𝑏 𝑝 𝑞 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-img 32098 . . 3 Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)
21breqi 4691 . 2 (⟨𝐴, 𝐵⟩Img𝐶 ↔ ⟨𝐴, 𝐵⟩(Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)𝐶)
3 opex 4962 . . . 4 𝐴, 𝐵⟩ ∈ V
4 brimg.3 . . . 4 𝐶 ∈ V
53, 4brco 5325 . . 3 (⟨𝐴, 𝐵⟩(Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)𝐶 ↔ ∃𝑎(⟨𝐴, 𝐵⟩Cart𝑎𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶))
6 brimg.1 . . . . . 6 𝐴 ∈ V
7 brimg.2 . . . . . 6 𝐵 ∈ V
8 vex 3234 . . . . . 6 𝑎 ∈ V
96, 7, 8brcart 32164 . . . . 5 (⟨𝐴, 𝐵⟩Cart𝑎𝑎 = (𝐴 × 𝐵))
109anbi1i 731 . . . 4 ((⟨𝐴, 𝐵⟩Cart𝑎𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶) ↔ (𝑎 = (𝐴 × 𝐵) ∧ 𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶))
1110exbii 1814 . . 3 (∃𝑎(⟨𝐴, 𝐵⟩Cart𝑎𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶) ↔ ∃𝑎(𝑎 = (𝐴 × 𝐵) ∧ 𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶))
126, 7xpex 7004 . . . 4 (𝐴 × 𝐵) ∈ V
13 breq1 4688 . . . 4 (𝑎 = (𝐴 × 𝐵) → (𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶 ↔ (𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶))
1412, 13ceqsexv 3273 . . 3 (∃𝑎(𝑎 = (𝐴 × 𝐵) ∧ 𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶) ↔ (𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶)
155, 11, 143bitri 286 . 2 (⟨𝐴, 𝐵⟩(Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)𝐶 ↔ (𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶)
1612, 4brimage 32158 . . 3 ((𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶𝐶 = (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)))
17 19.42v 1921 . . . . . . . 8 (∃𝑎(𝑏𝐵 ∧ (𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)) ↔ (𝑏𝐵 ∧ ∃𝑎(𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
18 anass 682 . . . . . . . . . . 11 (((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎𝐴𝑏𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
19 anass 682 . . . . . . . . . . . . 13 (((𝑎𝐴𝑏𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (𝑎𝐴 ∧ (𝑏𝐵𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
20 an12 855 . . . . . . . . . . . . 13 ((𝑎𝐴 ∧ (𝑏𝐵𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)) ↔ (𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
2119, 20bitri 264 . . . . . . . . . . . 12 (((𝑎𝐴𝑏𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
2221anbi2i 730 . . . . . . . . . . 11 ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎𝐴𝑏𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)) ↔ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))))
2318, 22bitri 264 . . . . . . . . . 10 (((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))))
24232exbii 1815 . . . . . . . . 9 (∃𝑝𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑝𝑎(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))))
25 excom 2082 . . . . . . . . 9 (∃𝑝𝑎(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))) ↔ ∃𝑎𝑝(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))))
26 opex 4962 . . . . . . . . . . 11 𝑎, 𝑏⟩ ∈ V
27 breq1 4688 . . . . . . . . . . . . 13 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥 ↔ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
2827anbi2d 740 . . . . . . . . . . . 12 (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
2928anbi2d 740 . . . . . . . . . . 11 (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)) ↔ (𝑏𝐵 ∧ (𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))))
3026, 29ceqsexv 3273 . . . . . . . . . 10 (∃𝑝(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))) ↔ (𝑏𝐵 ∧ (𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
3130exbii 1814 . . . . . . . . 9 (∃𝑎𝑝(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))) ↔ ∃𝑎(𝑏𝐵 ∧ (𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
3224, 25, 313bitri 286 . . . . . . . 8 (∃𝑝𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑎(𝑏𝐵 ∧ (𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
33 df-br 4686 . . . . . . . . . 10 (𝑏𝐴𝑥 ↔ ⟨𝑏, 𝑥⟩ ∈ 𝐴)
34 risset 3091 . . . . . . . . . . 11 (⟨𝑏, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑎𝐴 𝑎 = ⟨𝑏, 𝑥⟩)
35 vex 3234 . . . . . . . . . . . . . . . 16 𝑏 ∈ V
3635brres 5437 . . . . . . . . . . . . . . 15 (𝑎(1st ↾ (V × V))𝑏 ↔ (𝑎1st 𝑏𝑎 ∈ (V × V)))
37 df-br 4686 . . . . . . . . . . . . . . 15 (𝑎(1st ↾ (V × V))𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ (1st ↾ (V × V)))
38 ancom 465 . . . . . . . . . . . . . . 15 ((𝑎1st 𝑏𝑎 ∈ (V × V)) ↔ (𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏))
3936, 37, 383bitr3ri 291 . . . . . . . . . . . . . 14 ((𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏) ↔ ⟨𝑎, 𝑏⟩ ∈ (1st ↾ (V × V)))
4039anbi2i 730 . . . . . . . . . . . . 13 ((⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥 ∧ (𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏)) ↔ (⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥 ∧ ⟨𝑎, 𝑏⟩ ∈ (1st ↾ (V × V))))
41 elvv 5211 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (V × V) ↔ ∃𝑝𝑞 𝑎 = ⟨𝑝, 𝑞⟩)
4241anbi1i 731 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (V × V) ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ (∃𝑝𝑞 𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)))
43 anass 682 . . . . . . . . . . . . . . 15 (((𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏) ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (𝑎 ∈ (V × V) ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)))
44 ancom 465 . . . . . . . . . . . . . . . . . 18 ((𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑝 = 𝑏𝑞 = 𝑥)) ↔ ((𝑝 = 𝑏𝑞 = 𝑥) ∧ 𝑎 = ⟨𝑝, 𝑞⟩))
45 breq1 4688 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = ⟨𝑝, 𝑞⟩ → (𝑎1st 𝑏 ↔ ⟨𝑝, 𝑞⟩1st 𝑏))
46 opeq1 4433 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = ⟨𝑝, 𝑞⟩ → ⟨𝑎, 𝑏⟩ = ⟨⟨𝑝, 𝑞⟩, 𝑏⟩)
4746breq1d 4695 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = ⟨𝑝, 𝑞⟩ → (⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥 ↔ ⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥))
4845, 47anbi12d 747 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = ⟨𝑝, 𝑞⟩ → ((𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (⟨𝑝, 𝑞⟩1st 𝑏 ∧ ⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥)))
49 vex 3234 . . . . . . . . . . . . . . . . . . . . . . 23 𝑝 ∈ V
50 vex 3234 . . . . . . . . . . . . . . . . . . . . . . 23 𝑞 ∈ V
5149, 50br1steq 31796 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑝, 𝑞⟩1st 𝑏𝑏 = 𝑝)
52 equcom 1991 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝑝𝑝 = 𝑏)
5351, 52bitri 264 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑝, 𝑞⟩1st 𝑏𝑝 = 𝑏)
54 opex 4962 . . . . . . . . . . . . . . . . . . . . . . . 24 ⟨⟨𝑝, 𝑞⟩, 𝑏⟩ ∈ V
55 vex 3234 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑥 ∈ V
5654, 55brco 5325 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥 ↔ ∃𝑎(⟨⟨𝑝, 𝑞⟩, 𝑏⟩1st 𝑎𝑎2nd 𝑥))
57 opex 4962 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑝, 𝑞⟩ ∈ V
5857, 35br1steq 31796 . . . . . . . . . . . . . . . . . . . . . . . . 25 (⟨⟨𝑝, 𝑞⟩, 𝑏⟩1st 𝑎𝑎 = ⟨𝑝, 𝑞⟩)
5958anbi1i 731 . . . . . . . . . . . . . . . . . . . . . . . 24 ((⟨⟨𝑝, 𝑞⟩, 𝑏⟩1st 𝑎𝑎2nd 𝑥) ↔ (𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥))
6059exbii 1814 . . . . . . . . . . . . . . . . . . . . . . 23 (∃𝑎(⟨⟨𝑝, 𝑞⟩, 𝑏⟩1st 𝑎𝑎2nd 𝑥) ↔ ∃𝑎(𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥))
6156, 60bitri 264 . . . . . . . . . . . . . . . . . . . . . 22 (⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥 ↔ ∃𝑎(𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥))
62 breq1 4688 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = ⟨𝑝, 𝑞⟩ → (𝑎2nd 𝑥 ↔ ⟨𝑝, 𝑞⟩2nd 𝑥))
6357, 62ceqsexv 3273 . . . . . . . . . . . . . . . . . . . . . . 23 (∃𝑎(𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥) ↔ ⟨𝑝, 𝑞⟩2nd 𝑥)
6449, 50br2ndeq 31797 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑝, 𝑞⟩2nd 𝑥𝑥 = 𝑞)
6563, 64bitri 264 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑎(𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥) ↔ 𝑥 = 𝑞)
66 equcom 1991 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑞𝑞 = 𝑥)
6761, 65, 663bitri 286 . . . . . . . . . . . . . . . . . . . . 21 (⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥𝑞 = 𝑥)
6853, 67anbi12i 733 . . . . . . . . . . . . . . . . . . . 20 ((⟨𝑝, 𝑞⟩1st 𝑏 ∧ ⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (𝑝 = 𝑏𝑞 = 𝑥))
6948, 68syl6bb 276 . . . . . . . . . . . . . . . . . . 19 (𝑎 = ⟨𝑝, 𝑞⟩ → ((𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (𝑝 = 𝑏𝑞 = 𝑥)))
7069pm5.32i 670 . . . . . . . . . . . . . . . . . 18 ((𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ (𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑝 = 𝑏𝑞 = 𝑥)))
71 df-3an 1056 . . . . . . . . . . . . . . . . . 18 ((𝑝 = 𝑏𝑞 = 𝑥𝑎 = ⟨𝑝, 𝑞⟩) ↔ ((𝑝 = 𝑏𝑞 = 𝑥) ∧ 𝑎 = ⟨𝑝, 𝑞⟩))
7244, 70, 713bitr4i 292 . . . . . . . . . . . . . . . . 17 ((𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ (𝑝 = 𝑏𝑞 = 𝑥𝑎 = ⟨𝑝, 𝑞⟩))
73722exbii 1815 . . . . . . . . . . . . . . . 16 (∃𝑝𝑞(𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ ∃𝑝𝑞(𝑝 = 𝑏𝑞 = 𝑥𝑎 = ⟨𝑝, 𝑞⟩))
74 19.41vv 1918 . . . . . . . . . . . . . . . 16 (∃𝑝𝑞(𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ (∃𝑝𝑞 𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)))
75 opeq1 4433 . . . . . . . . . . . . . . . . . 18 (𝑝 = 𝑏 → ⟨𝑝, 𝑞⟩ = ⟨𝑏, 𝑞⟩)
7675eqeq2d 2661 . . . . . . . . . . . . . . . . 17 (𝑝 = 𝑏 → (𝑎 = ⟨𝑝, 𝑞⟩ ↔ 𝑎 = ⟨𝑏, 𝑞⟩))
77 opeq2 4434 . . . . . . . . . . . . . . . . . 18 (𝑞 = 𝑥 → ⟨𝑏, 𝑞⟩ = ⟨𝑏, 𝑥⟩)
7877eqeq2d 2661 . . . . . . . . . . . . . . . . 17 (𝑞 = 𝑥 → (𝑎 = ⟨𝑏, 𝑞⟩ ↔ 𝑎 = ⟨𝑏, 𝑥⟩))
7935, 55, 76, 78ceqsex2v 3276 . . . . . . . . . . . . . . . 16 (∃𝑝𝑞(𝑝 = 𝑏𝑞 = 𝑥𝑎 = ⟨𝑝, 𝑞⟩) ↔ 𝑎 = ⟨𝑏, 𝑥⟩)
8073, 74, 793bitr3ri 291 . . . . . . . . . . . . . . 15 (𝑎 = ⟨𝑏, 𝑥⟩ ↔ (∃𝑝𝑞 𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)))
8142, 43, 803bitr4ri 293 . . . . . . . . . . . . . 14 (𝑎 = ⟨𝑏, 𝑥⟩ ↔ ((𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏) ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥))
82 ancom 465 . . . . . . . . . . . . . 14 (((𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏) ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥 ∧ (𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏)))
8381, 82bitri 264 . . . . . . . . . . . . 13 (𝑎 = ⟨𝑏, 𝑥⟩ ↔ (⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥 ∧ (𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏)))
8455brres 5437 . . . . . . . . . . . . 13 (⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥 ↔ (⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥 ∧ ⟨𝑎, 𝑏⟩ ∈ (1st ↾ (V × V))))
8540, 83, 843bitr4i 292 . . . . . . . . . . . 12 (𝑎 = ⟨𝑏, 𝑥⟩ ↔ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)
8685rexbii 3070 . . . . . . . . . . 11 (∃𝑎𝐴 𝑎 = ⟨𝑏, 𝑥⟩ ↔ ∃𝑎𝐴𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)
8734, 86bitri 264 . . . . . . . . . 10 (⟨𝑏, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑎𝐴𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)
88 df-rex 2947 . . . . . . . . . 10 (∃𝑎𝐴𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥 ↔ ∃𝑎(𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
8933, 87, 883bitri 286 . . . . . . . . 9 (𝑏𝐴𝑥 ↔ ∃𝑎(𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
9089anbi2i 730 . . . . . . . 8 ((𝑏𝐵𝑏𝐴𝑥) ↔ (𝑏𝐵 ∧ ∃𝑎(𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
9117, 32, 903bitr4ri 293 . . . . . . 7 ((𝑏𝐵𝑏𝐴𝑥) ↔ ∃𝑝𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
9291exbii 1814 . . . . . 6 (∃𝑏(𝑏𝐵𝑏𝐴𝑥) ↔ ∃𝑏𝑝𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
9355elima2 5507 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ ∃𝑏(𝑏𝐵𝑏𝐴𝑥))
9455elima2 5507 . . . . . . 7 (𝑥 ∈ (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) ↔ ∃𝑝(𝑝 ∈ (𝐴 × 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
95 elxp 5165 . . . . . . . . . 10 (𝑝 ∈ (𝐴 × 𝐵) ↔ ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)))
9695anbi1i 731 . . . . . . . . 9 ((𝑝 ∈ (𝐴 × 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
97 19.41vv 1918 . . . . . . . . 9 (∃𝑎𝑏((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
9896, 97bitr4i 267 . . . . . . . 8 ((𝑝 ∈ (𝐴 × 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑎𝑏((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
9998exbii 1814 . . . . . . 7 (∃𝑝(𝑝 ∈ (𝐴 × 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑝𝑎𝑏((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
100 exrot3 2085 . . . . . . . 8 (∃𝑝𝑎𝑏((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑎𝑏𝑝((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
101 exrot3 2085 . . . . . . . 8 (∃𝑎𝑏𝑝((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑏𝑝𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
102100, 101bitri 264 . . . . . . 7 (∃𝑝𝑎𝑏((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑏𝑝𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
10394, 99, 1023bitri 286 . . . . . 6 (𝑥 ∈ (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) ↔ ∃𝑏𝑝𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
10492, 93, 1033bitr4ri 293 . . . . 5 (𝑥 ∈ (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
105104eqriv 2648 . . . 4 (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) = (𝐴𝐵)
106105eqeq2i 2663 . . 3 (𝐶 = (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) ↔ 𝐶 = (𝐴𝐵))
10716, 106bitri 264 . 2 ((𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶𝐶 = (𝐴𝐵))
1082, 15, 1073bitri 286 1 (⟨𝐴, 𝐵⟩Img𝐶𝐶 = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wrex 2942  Vcvv 3231  cop 4216   class class class wbr 4685   × cxp 5141  cres 5145  cima 5146  ccom 5147  1st c1st 7208  2nd c2nd 7209  Imagecimage 32072  Cartccart 32073  Imgcimg 32074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-symdif 3877  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-eprel 5058  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-1st 7210  df-2nd 7211  df-txp 32086  df-pprod 32087  df-image 32096  df-cart 32097  df-img 32098
This theorem is referenced by:  brapply  32170  dfrdg4  32183
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