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Theorem xpinpreima 29758
Description: Rewrite the cartesian product of two sets as the intersection of their preimage by 1st and 2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)
Assertion
Ref Expression
xpinpreima (𝐴 × 𝐵) = (((1st ↾ (V × V)) “ 𝐴) ∩ ((2nd ↾ (V × V)) “ 𝐵))

Proof of Theorem xpinpreima
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 inrab 3880 . 2 ({𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵}) = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
2 f1stres 7142 . . . . 5 (1st ↾ (V × V)):(V × V)⟶V
3 ffn 6007 . . . . 5 ((1st ↾ (V × V)):(V × V)⟶V → (1st ↾ (V × V)) Fn (V × V))
4 fncnvima2 6300 . . . . 5 ((1st ↾ (V × V)) Fn (V × V) → ((1st ↾ (V × V)) “ 𝐴) = {𝑟 ∈ (V × V) ∣ ((1st ↾ (V × V))‘𝑟) ∈ 𝐴})
52, 3, 4mp2b 10 . . . 4 ((1st ↾ (V × V)) “ 𝐴) = {𝑟 ∈ (V × V) ∣ ((1st ↾ (V × V))‘𝑟) ∈ 𝐴}
6 fvres 6169 . . . . . 6 (𝑟 ∈ (V × V) → ((1st ↾ (V × V))‘𝑟) = (1st𝑟))
76eleq1d 2683 . . . . 5 (𝑟 ∈ (V × V) → (((1st ↾ (V × V))‘𝑟) ∈ 𝐴 ↔ (1st𝑟) ∈ 𝐴))
87rabbiia 3176 . . . 4 {𝑟 ∈ (V × V) ∣ ((1st ↾ (V × V))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴}
95, 8eqtri 2643 . . 3 ((1st ↾ (V × V)) “ 𝐴) = {𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴}
10 f2ndres 7143 . . . . 5 (2nd ↾ (V × V)):(V × V)⟶V
11 ffn 6007 . . . . 5 ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V))
12 fncnvima2 6300 . . . . 5 ((2nd ↾ (V × V)) Fn (V × V) → ((2nd ↾ (V × V)) “ 𝐵) = {𝑟 ∈ (V × V) ∣ ((2nd ↾ (V × V))‘𝑟) ∈ 𝐵})
1310, 11, 12mp2b 10 . . . 4 ((2nd ↾ (V × V)) “ 𝐵) = {𝑟 ∈ (V × V) ∣ ((2nd ↾ (V × V))‘𝑟) ∈ 𝐵}
14 fvres 6169 . . . . . 6 (𝑟 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑟) = (2nd𝑟))
1514eleq1d 2683 . . . . 5 (𝑟 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑟) ∈ 𝐵 ↔ (2nd𝑟) ∈ 𝐵))
1615rabbiia 3176 . . . 4 {𝑟 ∈ (V × V) ∣ ((2nd ↾ (V × V))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵}
1713, 16eqtri 2643 . . 3 ((2nd ↾ (V × V)) “ 𝐵) = {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵}
189, 17ineq12i 3795 . 2 (((1st ↾ (V × V)) “ 𝐴) ∩ ((2nd ↾ (V × V)) “ 𝐵)) = ({𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵})
19 xp2 7155 . 2 (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
201, 18, 193eqtr4ri 2654 1 (𝐴 × 𝐵) = (((1st ↾ (V × V)) “ 𝐴) ∩ ((2nd ↾ (V × V)) “ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3189  cin 3558   × cxp 5077  ccnv 5078  cres 5081  cima 5082   Fn wfn 5847  wf 5848  cfv 5852  1st c1st 7118  2nd c2nd 7119
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 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-1st 7120  df-2nd 7121
This theorem is referenced by: (None)
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