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Theorem xpinpreima 33658
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 4305 . 2 ({𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵}) = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
2 f1stres 8018 . . . . 5 (1st ↾ (V × V)):(V × V)⟶V
3 ffn 6723 . . . . 5 ((1st ↾ (V × V)):(V × V)⟶V → (1st ↾ (V × V)) Fn (V × V))
4 fncnvima2 7069 . . . . 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 6915 . . . . . 6 (𝑟 ∈ (V × V) → ((1st ↾ (V × V))‘𝑟) = (1st𝑟))
76eleq1d 2810 . . . . 5 (𝑟 ∈ (V × V) → (((1st ↾ (V × V))‘𝑟) ∈ 𝐴 ↔ (1st𝑟) ∈ 𝐴))
87rabbiia 3422 . . . 4 {𝑟 ∈ (V × V) ∣ ((1st ↾ (V × V))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴}
95, 8eqtri 2753 . . 3 ((1st ↾ (V × V)) “ 𝐴) = {𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴}
10 f2ndres 8019 . . . . 5 (2nd ↾ (V × V)):(V × V)⟶V
11 ffn 6723 . . . . 5 ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V))
12 fncnvima2 7069 . . . . 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 6915 . . . . . 6 (𝑟 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑟) = (2nd𝑟))
1514eleq1d 2810 . . . . 5 (𝑟 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑟) ∈ 𝐵 ↔ (2nd𝑟) ∈ 𝐵))
1615rabbiia 3422 . . . 4 {𝑟 ∈ (V × V) ∣ ((2nd ↾ (V × V))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵}
1713, 16eqtri 2753 . . 3 ((2nd ↾ (V × V)) “ 𝐵) = {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵}
189, 17ineq12i 4208 . 2 (((1st ↾ (V × V)) “ 𝐴) ∩ ((2nd ↾ (V × V)) “ 𝐵)) = ({𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵})
19 xp2 8031 . 2 (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
201, 18, 193eqtr4ri 2764 1 (𝐴 × 𝐵) = (((1st ↾ (V × V)) “ 𝐴) ∩ ((2nd ↾ (V × V)) “ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wcel 2098  {crab 3418  Vcvv 3461  cin 3943   × cxp 5676  ccnv 5677  cres 5680  cima 5681   Fn wfn 6544  wf 6545  cfv 6549  1st c1st 7992  2nd c2nd 7993
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-1st 7994  df-2nd 7995
This theorem is referenced by: (None)
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