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

Theorem brimg 33283
 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⟩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 33212 . . 3 Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)
21breqi 5068 . 2 (⟨𝐴, 𝐵⟩Img𝐶 ↔ ⟨𝐴, 𝐵⟩(Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)𝐶)
3 opex 5352 . . . 4 𝐴, 𝐵⟩ ∈ V
4 brimg.3 . . . 4 𝐶 ∈ V
53, 4brco 5739 . . 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 3502 . . . . . 6 𝑎 ∈ V
96, 7, 8brcart 33278 . . . . 5 (⟨𝐴, 𝐵⟩Cart𝑎𝑎 = (𝐴 × 𝐵))
109anbi1i 623 . . . 4 ((⟨𝐴, 𝐵⟩Cart𝑎𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶) ↔ (𝑎 = (𝐴 × 𝐵) ∧ 𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶))
1110exbii 1841 . . 3 (∃𝑎(⟨𝐴, 𝐵⟩Cart𝑎𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶) ↔ ∃𝑎(𝑎 = (𝐴 × 𝐵) ∧ 𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶))
126, 7xpex 7468 . . . 4 (𝐴 × 𝐵) ∈ V
13 breq1 5065 . . . 4 (𝑎 = (𝐴 × 𝐵) → (𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶 ↔ (𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶))
1412, 13ceqsexv 3546 . . 3 (∃𝑎(𝑎 = (𝐴 × 𝐵) ∧ 𝑎Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶) ↔ (𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶)
155, 11, 143bitri 298 . 2 (⟨𝐴, 𝐵⟩(Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)𝐶 ↔ (𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶)
1612, 4brimage 33272 . . 3 ((𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶𝐶 = (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)))
17 19.42v 1947 . . . . . . . 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 469 . . . . . . . . . . 11 (((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎𝐴𝑏𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
19 an21 640 . . . . . . . . . . . 12 (((𝑎𝐴𝑏𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
2019anbi2i 622 . . . . . . . . . . 11 ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎𝐴𝑏𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)) ↔ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))))
2118, 20bitri 276 . . . . . . . . . 10 (((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))))
22212exbii 1842 . . . . . . . . 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))))
23 excom 2159 . . . . . . . . 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))))
24 opex 5352 . . . . . . . . . . 11 𝑎, 𝑏⟩ ∈ V
25 breq1 5065 . . . . . . . . . . . . 13 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥 ↔ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
2625anbi2d 628 . . . . . . . . . . . 12 (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
2726anbi2d 628 . . . . . . . . . . 11 (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)) ↔ (𝑏𝐵 ∧ (𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))))
2824, 27ceqsexv 3546 . . . . . . . . . 10 (∃𝑝(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))) ↔ (𝑏𝐵 ∧ (𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
2928exbii 1841 . . . . . . . . 9 (∃𝑎𝑝(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑏𝐵 ∧ (𝑎𝐴𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))) ↔ ∃𝑎(𝑏𝐵 ∧ (𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
3022, 23, 293bitri 298 . . . . . . . 8 (∃𝑝𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑎(𝑏𝐵 ∧ (𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
31 df-br 5063 . . . . . . . . . 10 (𝑏𝐴𝑥 ↔ ⟨𝑏, 𝑥⟩ ∈ 𝐴)
32 risset 3271 . . . . . . . . . . 11 (⟨𝑏, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑎𝐴 𝑎 = ⟨𝑏, 𝑥⟩)
33 vex 3502 . . . . . . . . . . . . . . . 16 𝑏 ∈ V
3433brresi 5860 . . . . . . . . . . . . . . 15 (𝑎(1st ↾ (V × V))𝑏 ↔ (𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏))
35 df-br 5063 . . . . . . . . . . . . . . 15 (𝑎(1st ↾ (V × V))𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ (1st ↾ (V × V)))
3634, 35bitr3i 278 . . . . . . . . . . . . . 14 ((𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏) ↔ ⟨𝑎, 𝑏⟩ ∈ (1st ↾ (V × V)))
3736anbi1i 623 . . . . . . . . . . . . 13 (((𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏) ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ (1st ↾ (V × V)) ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥))
38 elvv 5624 . . . . . . . . . . . . . . 15 (𝑎 ∈ (V × V) ↔ ∃𝑝𝑞 𝑎 = ⟨𝑝, 𝑞⟩)
3938anbi1i 623 . . . . . . . . . . . . . 14 ((𝑎 ∈ (V × V) ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ (∃𝑝𝑞 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 = 𝑏𝑞 = 𝑥)) ↔ ((𝑝 = 𝑏𝑞 = 𝑥) ∧ 𝑎 = ⟨𝑝, 𝑞⟩))
42 breq1 5065 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = ⟨𝑝, 𝑞⟩ → (𝑎1st 𝑏 ↔ ⟨𝑝, 𝑞⟩1st 𝑏))
43 opeq1 4801 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = ⟨𝑝, 𝑞⟩ → ⟨𝑎, 𝑏⟩ = ⟨⟨𝑝, 𝑞⟩, 𝑏⟩)
4443breq1d 5072 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = ⟨𝑝, 𝑞⟩ → (⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥 ↔ ⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥))
4542, 44anbi12d 630 . . . . . . . . . . . . . . . . . . 19 (𝑎 = ⟨𝑝, 𝑞⟩ → ((𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (⟨𝑝, 𝑞⟩1st 𝑏 ∧ ⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥)))
46 vex 3502 . . . . . . . . . . . . . . . . . . . . . 22 𝑝 ∈ V
47 vex 3502 . . . . . . . . . . . . . . . . . . . . . 22 𝑞 ∈ V
4846, 47br1steq 32899 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑝, 𝑞⟩1st 𝑏𝑏 = 𝑝)
49 equcom 2018 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = 𝑝𝑝 = 𝑏)
5048, 49bitri 276 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑝, 𝑞⟩1st 𝑏𝑝 = 𝑏)
51 opex 5352 . . . . . . . . . . . . . . . . . . . . . . 23 ⟨⟨𝑝, 𝑞⟩, 𝑏⟩ ∈ V
52 vex 3502 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥 ∈ V
5351, 52brco 5739 . . . . . . . . . . . . . . . . . . . . . 22 (⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥 ↔ ∃𝑎(⟨⟨𝑝, 𝑞⟩, 𝑏⟩1st 𝑎𝑎2nd 𝑥))
54 opex 5352 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑝, 𝑞⟩ ∈ V
5554, 33br1steq 32899 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨⟨𝑝, 𝑞⟩, 𝑏⟩1st 𝑎𝑎 = ⟨𝑝, 𝑞⟩)
5655anbi1i 623 . . . . . . . . . . . . . . . . . . . . . . 23 ((⟨⟨𝑝, 𝑞⟩, 𝑏⟩1st 𝑎𝑎2nd 𝑥) ↔ (𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥))
5756exbii 1841 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑎(⟨⟨𝑝, 𝑞⟩, 𝑏⟩1st 𝑎𝑎2nd 𝑥) ↔ ∃𝑎(𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥))
5853, 57bitri 276 . . . . . . . . . . . . . . . . . . . . 21 (⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥 ↔ ∃𝑎(𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥))
59 breq1 5065 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = ⟨𝑝, 𝑞⟩ → (𝑎2nd 𝑥 ↔ ⟨𝑝, 𝑞⟩2nd 𝑥))
6054, 59ceqsexv 3546 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑎(𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥) ↔ ⟨𝑝, 𝑞⟩2nd 𝑥)
6146, 47br2ndeq 32900 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑝, 𝑞⟩2nd 𝑥𝑥 = 𝑞)
6260, 61bitri 276 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑎(𝑎 = ⟨𝑝, 𝑞⟩ ∧ 𝑎2nd 𝑥) ↔ 𝑥 = 𝑞)
63 equcom 2018 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑞𝑞 = 𝑥)
6458, 62, 633bitri 298 . . . . . . . . . . . . . . . . . . . 20 (⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥𝑞 = 𝑥)
6550, 64anbi12i 626 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑝, 𝑞⟩1st 𝑏 ∧ ⟨⟨𝑝, 𝑞⟩, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (𝑝 = 𝑏𝑞 = 𝑥))
6645, 65syl6bb 288 . . . . . . . . . . . . . . . . . 18 (𝑎 = ⟨𝑝, 𝑞⟩ → ((𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥) ↔ (𝑝 = 𝑏𝑞 = 𝑥)))
6766pm5.32i 575 . . . . . . . . . . . . . . . . 17 ((𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ (𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑝 = 𝑏𝑞 = 𝑥)))
68 df-3an 1083 . . . . . . . . . . . . . . . . 17 ((𝑝 = 𝑏𝑞 = 𝑥𝑎 = ⟨𝑝, 𝑞⟩) ↔ ((𝑝 = 𝑏𝑞 = 𝑥) ∧ 𝑎 = ⟨𝑝, 𝑞⟩))
6941, 67, 683bitr4i 304 . . . . . . . . . . . . . . . 16 ((𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ (𝑝 = 𝑏𝑞 = 𝑥𝑎 = ⟨𝑝, 𝑞⟩))
70692exbii 1842 . . . . . . . . . . . . . . 15 (∃𝑝𝑞(𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ ∃𝑝𝑞(𝑝 = 𝑏𝑞 = 𝑥𝑎 = ⟨𝑝, 𝑞⟩))
71 19.41vv 1944 . . . . . . . . . . . . . . 15 (∃𝑝𝑞(𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)) ↔ (∃𝑝𝑞 𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)))
72 opeq1 4801 . . . . . . . . . . . . . . . . 17 (𝑝 = 𝑏 → ⟨𝑝, 𝑞⟩ = ⟨𝑏, 𝑞⟩)
7372eqeq2d 2835 . . . . . . . . . . . . . . . 16 (𝑝 = 𝑏 → (𝑎 = ⟨𝑝, 𝑞⟩ ↔ 𝑎 = ⟨𝑏, 𝑞⟩))
74 opeq2 4802 . . . . . . . . . . . . . . . . 17 (𝑞 = 𝑥 → ⟨𝑏, 𝑞⟩ = ⟨𝑏, 𝑥⟩)
7574eqeq2d 2835 . . . . . . . . . . . . . . . 16 (𝑞 = 𝑥 → (𝑎 = ⟨𝑏, 𝑞⟩ ↔ 𝑎 = ⟨𝑏, 𝑥⟩))
7633, 52, 73, 75ceqsex2v 3549 . . . . . . . . . . . . . . 15 (∃𝑝𝑞(𝑝 = 𝑏𝑞 = 𝑥𝑎 = ⟨𝑝, 𝑞⟩) ↔ 𝑎 = ⟨𝑏, 𝑥⟩)
7770, 71, 763bitr3ri 303 . . . . . . . . . . . . . 14 (𝑎 = ⟨𝑏, 𝑥⟩ ↔ (∃𝑝𝑞 𝑎 = ⟨𝑝, 𝑞⟩ ∧ (𝑎1st 𝑏 ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥)))
7839, 40, 773bitr4ri 305 . . . . . . . . . . . . 13 (𝑎 = ⟨𝑏, 𝑥⟩ ↔ ((𝑎 ∈ (V × V) ∧ 𝑎1st 𝑏) ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥))
7952brresi 5860 . . . . . . . . . . . . 13 (⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥 ↔ (⟨𝑎, 𝑏⟩ ∈ (1st ↾ (V × V)) ∧ ⟨𝑎, 𝑏⟩(2nd ∘ 1st )𝑥))
8037, 78, 793bitr4i 304 . . . . . . . . . . . 12 (𝑎 = ⟨𝑏, 𝑥⟩ ↔ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)
8180rexbii 3251 . . . . . . . . . . 11 (∃𝑎𝐴 𝑎 = ⟨𝑏, 𝑥⟩ ↔ ∃𝑎𝐴𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)
8232, 81bitri 276 . . . . . . . . . 10 (⟨𝑏, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑎𝐴𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)
83 df-rex 3148 . . . . . . . . . 10 (∃𝑎𝐴𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥 ↔ ∃𝑎(𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
8431, 82, 833bitri 298 . . . . . . . . 9 (𝑏𝐴𝑥 ↔ ∃𝑎(𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
8584anbi2i 622 . . . . . . . 8 ((𝑏𝐵𝑏𝐴𝑥) ↔ (𝑏𝐵 ∧ ∃𝑎(𝑎𝐴 ∧ ⟨𝑎, 𝑏⟩((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥)))
8617, 30, 853bitr4ri 305 . . . . . . 7 ((𝑏𝐵𝑏𝐴𝑥) ↔ ∃𝑝𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
8786exbii 1841 . . . . . 6 (∃𝑏(𝑏𝐵𝑏𝐴𝑥) ↔ ∃𝑏𝑝𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
8852elima2 5932 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ ∃𝑏(𝑏𝐵𝑏𝐴𝑥))
8952elima2 5932 . . . . . . 7 (𝑥 ∈ (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) ↔ ∃𝑝(𝑝 ∈ (𝐴 × 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
90 elxp 5576 . . . . . . . . . 10 (𝑝 ∈ (𝐴 × 𝐵) ↔ ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)))
9190anbi1i 623 . . . . . . . . 9 ((𝑝 ∈ (𝐴 × 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ (∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
92 19.41vv 1944 . . . . . . . . 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))
9391, 92bitr4i 279 . . . . . . . 8 ((𝑝 ∈ (𝐴 × 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑎𝑏((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
9493exbii 1841 . . . . . . 7 (∃𝑝(𝑝 ∈ (𝐴 × 𝐵) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥) ↔ ∃𝑝𝑎𝑏((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
95 exrot3 2162 . . . . . . . 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))
96 exrot3 2162 . . . . . . . 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))
9795, 96bitri 276 . . . . . . 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))
9889, 94, 973bitri 298 . . . . . 6 (𝑥 ∈ (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) ↔ ∃𝑏𝑝𝑎((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑝((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝑥))
9987, 88, 983bitr4ri 305 . . . . 5 (𝑥 ∈ (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
10099eqriv 2821 . . . 4 (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) = (𝐴𝐵)
101100eqeq2i 2837 . . 3 (𝐶 = (((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) “ (𝐴 × 𝐵)) ↔ 𝐶 = (𝐴𝐵))
10216, 101bitri 276 . 2 ((𝐴 × 𝐵)Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V)))𝐶𝐶 = (𝐴𝐵))
1032, 15, 1023bitri 298 1 (⟨𝐴, 𝐵⟩Img𝐶𝐶 = (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∧ wa 396   ∧ w3a 1081   = wceq 1530  ∃wex 1773   ∈ wcel 2106  ∃wrex 3143  Vcvv 3499  ⟨cop 4569   class class class wbr 5062   × cxp 5551   ↾ cres 5555   “ cima 5556   ∘ ccom 5557  1st c1st 7681  2nd c2nd 7682  Imagecimage 33186  Cartccart 33187  Imgcimg 33188 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-symdif 4222  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-eprel 5463  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fo 6357  df-fv 6359  df-1st 7683  df-2nd 7684  df-txp 33200  df-pprod 33201  df-image 33210  df-cart 33211  df-img 33212 This theorem is referenced by:  brapply  33284  dfrdg4  33297
 Copyright terms: Public domain W3C validator