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Theorem xpinpreima 30277
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 4100 . 2 ({𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵}) = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
2 f1stres 7422 . . . . 5 (1st ↾ (V × V)):(V × V)⟶V
3 ffn 6256 . . . . 5 ((1st ↾ (V × V)):(V × V)⟶V → (1st ↾ (V × V)) Fn (V × V))
4 fncnvima2 6561 . . . . 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 6427 . . . . . 6 (𝑟 ∈ (V × V) → ((1st ↾ (V × V))‘𝑟) = (1st𝑟))
76eleq1d 2870 . . . . 5 (𝑟 ∈ (V × V) → (((1st ↾ (V × V))‘𝑟) ∈ 𝐴 ↔ (1st𝑟) ∈ 𝐴))
87rabbiia 3374 . . . 4 {𝑟 ∈ (V × V) ∣ ((1st ↾ (V × V))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴}
95, 8eqtri 2828 . . 3 ((1st ↾ (V × V)) “ 𝐴) = {𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴}
10 f2ndres 7423 . . . . 5 (2nd ↾ (V × V)):(V × V)⟶V
11 ffn 6256 . . . . 5 ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V))
12 fncnvima2 6561 . . . . 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 6427 . . . . . 6 (𝑟 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑟) = (2nd𝑟))
1514eleq1d 2870 . . . . 5 (𝑟 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑟) ∈ 𝐵 ↔ (2nd𝑟) ∈ 𝐵))
1615rabbiia 3374 . . . 4 {𝑟 ∈ (V × V) ∣ ((2nd ↾ (V × V))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵}
1713, 16eqtri 2828 . . 3 ((2nd ↾ (V × V)) “ 𝐵) = {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵}
189, 17ineq12i 4011 . 2 (((1st ↾ (V × V)) “ 𝐴) ∩ ((2nd ↾ (V × V)) “ 𝐵)) = ({𝑟 ∈ (V × V) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd𝑟) ∈ 𝐵})
19 xp2 7435 . 2 (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
201, 18, 193eqtr4ri 2839 1 (𝐴 × 𝐵) = (((1st ↾ (V × V)) “ 𝐴) ∩ ((2nd ↾ (V × V)) “ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1637  wcel 2156  {crab 3100  Vcvv 3391  cin 3768   × cxp 5309  ccnv 5310  cres 5313  cima 5314   Fn wfn 6096  wf 6097  cfv 6101  1st c1st 7396  2nd c2nd 7397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-1st 7398  df-2nd 7399
This theorem is referenced by: (None)
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