Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brimg Structured version   Visualization version   GIF version

Theorem brimg 31683
Description: The binary relationship 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 31611 . . 3 Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)
21breqi 4619 . 2 (⟨𝐴, 𝐵⟩Img𝐶 ↔ ⟨𝐴, 𝐵⟩(Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)𝐶)
3 opex 4893 . . . 4 𝐴, 𝐵⟩ ∈ V
4 brimg.3 . . . 4 𝐶 ∈ V
53, 4brco 5252 . . 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 3189 . . . . . 6 𝑎 ∈ V
96, 7, 8brcart 31678 . . . . 5 (⟨𝐴, 𝐵⟩Cart𝑎𝑎 = (𝐴 × 𝐵))
109anbi1i 730 . . . 4 ((⟨𝐴, 𝐵⟩Cart𝑎𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶) ↔ (𝑎 = (𝐴 × 𝐵) ∧ 𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶))
1110exbii 1771 . . 3 (∃𝑎(⟨𝐴, 𝐵⟩Cart𝑎𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶) ↔ ∃𝑎(𝑎 = (𝐴 × 𝐵) ∧ 𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶))
126, 7xpex 6915 . . . 4 (𝐴 × 𝐵) ∈ V
13 breq1 4616 . . . 4 (𝑎 = (𝐴 × 𝐵) → (𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶 ↔ (𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶))
1412, 13ceqsexv 3228 . . 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 31672 . . 3 ((𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶𝐶 = (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)))
17 19.42v 1915 . . . . . . . 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 680 . . . . . . . . . . 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 680 . . . . . . . . . . . . 13 (((𝑎𝐴𝑏𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (𝑎𝐴 ∧ (𝑏𝐵𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
20 an12 837 . . . . . . . . . . . . 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 729 . . . . . . . . . . 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 1772 . . . . . . . . 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 2039 . . . . . . . . 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 4893 . . . . . . . . . . 11 𝑎, 𝑏⟩ ∈ V
27 breq1 4616 . . . . . . . . . . . . 13 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥 ↔ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
2827anbi2d 739 . . . . . . . . . . . 12 (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
2928anbi2d 739 . . . . . . . . . . 11 (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)) ↔ (𝑏𝐵 ∧ (𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))))
3026, 29ceqsexv 3228 . . . . . . . . . 10 (∃𝑝(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))) ↔ (𝑏𝐵 ∧ (𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
3130exbii 1771 . . . . . . . . 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 4614 . . . . . . . . . 10 (𝑏𝐴𝑥 ↔ ⟨𝑏, 𝑥⟩ ∈ 𝐴)
34 risset 3055 . . . . . . . . . . 11 (⟨𝑏, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑎𝐴 𝑎 = ⟨𝑏, 𝑥⟩)
35 vex 3189 . . . . . . . . . . . . . . . 16 𝑏 ∈ V
3635brres 5362 . . . . . . . . . . . . . . 15 (𝑎(1st ↾ (V × V))𝑏 ↔ (𝑎1st 𝑏𝑎 ∈ (V × V)))
37 df-br 4614 . . . . . . . . . . . . . . 15 (𝑎(1st ↾ (V × V))𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ (1st ↾ (V × V)))
38 ancom 466 . . . . . . . . . . . . . . 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 729 . . . . . . . . . . . . 13 ((⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥 ∧ (𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏)) ↔ (⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥 ∧ ⟨𝑎, 𝑏⟩ ∈ (1st ↾ (V × V))))
41 elvv 5138 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (V × V) ↔ ∃𝑝𝑞 𝑎 = ⟨𝑝, 𝑞⟩)
4241anbi1i 730 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (V × V) ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ (∃𝑝𝑞 𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)))
43 anass 680 . . . . . . . . . . . . . . 15 (((𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏) ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (𝑎 ∈ (V × V) ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)))
44 ancom 466 . . . . . . . . . . . . . . . . . 18 ((𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑝 = 𝑏𝑞 = 𝑥)) ↔ ((𝑝 = 𝑏𝑞 = 𝑥) ∧ 𝑎 = ⟨𝑝, 𝑞⟩))
45 breq1 4616 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = ⟨𝑝, 𝑞⟩ → (𝑎1st 𝑏 ↔ ⟨𝑝, 𝑞⟩1st 𝑏))
46 opeq1 4370 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = ⟨𝑝, 𝑞⟩ → ⟨𝑎, 𝑏⟩ = ⟨⟨𝑝, 𝑞⟩, 𝑏⟩)
4746breq1d 4623 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = ⟨𝑝, 𝑞⟩ → (⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥 ↔ ⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥))
4845, 47anbi12d 746 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = ⟨𝑝, 𝑞⟩ → ((𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (⟨𝑝, 𝑞⟩1st 𝑏 ∧ ⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥)))
49 vex 3189 . . . . . . . . . . . . . . . . . . . . . . 23 𝑝 ∈ V
50 vex 3189 . . . . . . . . . . . . . . . . . . . . . . 23 𝑞 ∈ V
5149, 50, 35br1steq 31371 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑝, 𝑞⟩1st 𝑏𝑏 = 𝑝)
52 equcom 1942 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝑝𝑝 = 𝑏)
5351, 52bitri 264 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑝, 𝑞⟩1st 𝑏𝑝 = 𝑏)
54 opex 4893 . . . . . . . . . . . . . . . . . . . . . . . 24 ⟨⟨𝑝, 𝑞⟩, 𝑏⟩ ∈ V
55 vex 3189 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑥 ∈ V
5654, 55brco 5252 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥 ↔ ∃𝑎(⟨⟨𝑝, 𝑞⟩, 𝑏⟩1st 𝑎𝑎2nd 𝑥))
57 opex 4893 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑝, 𝑞⟩ ∈ V
5857, 35, 8br1steq 31371 . . . . . . . . . . . . . . . . . . . . . . . . 25 (⟨⟨𝑝, 𝑞⟩, 𝑏⟩1st 𝑎𝑎 = ⟨𝑝, 𝑞⟩)
5958anbi1i 730 . . . . . . . . . . . . . . . . . . . . . . . 24 ((⟨⟨𝑝, 𝑞⟩, 𝑏⟩1st 𝑎𝑎2nd 𝑥) ↔ (𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥))
6059exbii 1771 . . . . . . . . . . . . . . . . . . . . . . 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 4616 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = ⟨𝑝, 𝑞⟩ → (𝑎2nd 𝑥 ↔ ⟨𝑝, 𝑞⟩2nd 𝑥))
6357, 62ceqsexv 3228 . . . . . . . . . . . . . . . . . . . . . . 23 (∃𝑎(𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥) ↔ ⟨𝑝, 𝑞⟩2nd 𝑥)
6449, 50, 55br2ndeq 31372 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑝, 𝑞⟩2nd 𝑥𝑥 = 𝑞)
6563, 64bitri 264 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑎(𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥) ↔ 𝑥 = 𝑞)
66 equcom 1942 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑞𝑞 = 𝑥)
6761, 65, 663bitri 286 . . . . . . . . . . . . . . . . . . . . 21 (⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥𝑞 = 𝑥)
6853, 67anbi12i 732 . . . . . . . . . . . . . . . . . . . 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 668 . . . . . . . . . . . . . . . . . 18 ((𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ (𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑝 = 𝑏𝑞 = 𝑥)))
71 df-3an 1038 . . . . . . . . . . . . . . . . . 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 1772 . . . . . . . . . . . . . . . 16 (∃𝑝𝑞(𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ ∃𝑝𝑞(𝑝 = 𝑏𝑞 = 𝑥𝑎 = ⟨𝑝, 𝑞⟩))
74 19.41vv 1912 . . . . . . . . . . . . . . . 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 4370 . . . . . . . . . . . . . . . . . 18 (𝑝 = 𝑏 → ⟨𝑝, 𝑞⟩ = ⟨𝑏, 𝑞⟩)
7675eqeq2d 2631 . . . . . . . . . . . . . . . . 17 (𝑝 = 𝑏 → (𝑎 = ⟨𝑝, 𝑞⟩ ↔ 𝑎 = ⟨𝑏, 𝑞⟩))
77 opeq2 4371 . . . . . . . . . . . . . . . . . 18 (𝑞 = 𝑥 → ⟨𝑏, 𝑞⟩ = ⟨𝑏, 𝑥⟩)
7877eqeq2d 2631 . . . . . . . . . . . . . . . . 17 (𝑞 = 𝑥 → (𝑎 = ⟨𝑏, 𝑞⟩ ↔ 𝑎 = ⟨𝑏, 𝑥⟩))
7935, 55, 76, 78ceqsex2v 3231 . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . 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 5362 . . . . . . . . . . . . 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 3034 . . . . . . . . . . 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 2913 . . . . . . . . . 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 729 . . . . . . . 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 1771 . . . . . 6 (∃𝑏(𝑏𝐵𝑏𝐴𝑥) ↔ ∃𝑏𝑝𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
9355elima2 5431 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ ∃𝑏(𝑏𝐵𝑏𝐴𝑥))
9455elima2 5431 . . . . . . 7 (𝑥 ∈ (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) ↔ ∃𝑝(𝑝 ∈ (𝐴 × 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
95 elxp 5091 . . . . . . . . . 10 (𝑝 ∈ (𝐴 × 𝐵) ↔ ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)))
9695anbi1i 730 . . . . . . . . 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 1912 . . . . . . . . 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 1771 . . . . . . 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 2042 . . . . . . . 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 2042 . . . . . . . 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 2618 . . . 4 (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) = (𝐴𝐵)
106105eqeq2i 2633 . . 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 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wrex 2908  Vcvv 3186  cop 4154   class class class wbr 4613   × cxp 5072  cres 5076  cima 5077  ccom 5078  1st c1st 7111  2nd c2nd 7112  Imagecimage 31585  Cartccart 31586  Imgcimg 31587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-symdif 3822  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-eprel 4985  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-1st 7113  df-2nd 7114  df-txp 31599  df-pprod 31600  df-image 31609  df-cart 31610  df-img 31611
This theorem is referenced by:  brapply  31684  dfrdg4  31697
  Copyright terms: Public domain W3C validator