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Theorem xpinpreima 31144
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 4275 . 2 ({𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵}) = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
2 f1stres 7707 . . . . 5 (1st ↾ (V × V)):(V × V)⟶V
3 ffn 6509 . . . . 5 ((1st ↾ (V × V)):(V × V)⟶V → (1st ↾ (V × V)) Fn (V × V))
4 fncnvima2 6826 . . . . 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 6684 . . . . . 6 (𝑟 ∈ (V × V) → ((1st ↾ (V × V))‘𝑟) = (1st𝑟))
76eleq1d 2897 . . . . 5 (𝑟 ∈ (V × V) → (((1st ↾ (V × V))‘𝑟) ∈ 𝐴 ↔ (1st𝑟) ∈ 𝐴))
87rabbiia 3473 . . . 4 {𝑟 ∈ (V × V) ∣ ((1st ↾ (V × V))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴}
95, 8eqtri 2844 . . 3 ((1st ↾ (V × V)) “ 𝐴) = {𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴}
10 f2ndres 7708 . . . . 5 (2nd ↾ (V × V)):(V × V)⟶V
11 ffn 6509 . . . . 5 ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V))
12 fncnvima2 6826 . . . . 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 6684 . . . . . 6 (𝑟 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑟) = (2nd𝑟))
1514eleq1d 2897 . . . . 5 (𝑟 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑟) ∈ 𝐵 ↔ (2nd𝑟) ∈ 𝐵))
1615rabbiia 3473 . . . 4 {𝑟 ∈ (V × V) ∣ ((2nd ↾ (V × V))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵}
1713, 16eqtri 2844 . . 3 ((2nd ↾ (V × V)) “ 𝐵) = {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵}
189, 17ineq12i 4187 . 2 (((1st ↾ (V × V)) “ 𝐴) ∩ ((2nd ↾ (V × V)) “ 𝐵)) = ({𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵})
19 xp2 7720 . 2 (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
201, 18, 193eqtr4ri 2855 1 (𝐴 × 𝐵) = (((1st ↾ (V × V)) “ 𝐴) ∩ ((2nd ↾ (V × V)) “ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1533  wcel 2110  {crab 3142  Vcvv 3495  cin 3935   × cxp 5548  ccnv 5549  cres 5552  cima 5553   Fn wfn 6345  wf 6346  cfv 6350  1st c1st 7681  2nd c2nd 7682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-1st 7683  df-2nd 7684
This theorem is referenced by: (None)
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