Theorem xpinpreima 33937

Description: Rewrite the cartesian product of two sets as the intersection of their preimage by 1^st and 2^nd, 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.

Step	Hyp	Ref	Expression
1		inrab 4291	. 2 ⊢ ({𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵}) = {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)}
2		f1stres 8012	. . . . 5 ⊢ (1st ↾ (V × V)):(V × V)⟶V
3		ffn 6706	. . . . 5 ⊢ ((1st ↾ (V × V)):(V × V)⟶V → (1st ↾ (V × V)) Fn (V × V))
4		fncnvima2 7051	. . . . 5 ⊢ ((1st ↾ (V × V)) Fn (V × V) → (◡(1st ↾ (V × V)) “ 𝐴) = {𝑟 ∈ (V × V) ∣ ((1st ↾ (V × V))‘𝑟) ∈ 𝐴})
5	2, 3, 4	mp2b 10	. . . 4 ⊢ (◡(1st ↾ (V × V)) “ 𝐴) = {𝑟 ∈ (V × V) ∣ ((1st ↾ (V × V))‘𝑟) ∈ 𝐴}
6		fvres 6895	. . . . . 6 ⊢ (𝑟 ∈ (V × V) → ((1st ↾ (V × V))‘𝑟) = (1st ‘𝑟))
7	6	eleq1d 2819	. . . . 5 ⊢ (𝑟 ∈ (V × V) → (((1st ↾ (V × V))‘𝑟) ∈ 𝐴 ↔ (1st ‘𝑟) ∈ 𝐴))
8	7	rabbiia 3419	. . . 4 ⊢ {𝑟 ∈ (V × V) ∣ ((1st ↾ (V × V))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴}
9	5, 8	eqtri 2758	. . 3 ⊢ (◡(1st ↾ (V × V)) “ 𝐴) = {𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴}
10		f2ndres 8013	. . . . 5 ⊢ (2nd ↾ (V × V)):(V × V)⟶V
11		ffn 6706	. . . . 5 ⊢ ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V))
12		fncnvima2 7051	. . . . 5 ⊢ ((2nd ↾ (V × V)) Fn (V × V) → (◡(2nd ↾ (V × V)) “ 𝐵) = {𝑟 ∈ (V × V) ∣ ((2nd ↾ (V × V))‘𝑟) ∈ 𝐵})
13	10, 11, 12	mp2b 10	. . . 4 ⊢ (◡(2nd ↾ (V × V)) “ 𝐵) = {𝑟 ∈ (V × V) ∣ ((2nd ↾ (V × V))‘𝑟) ∈ 𝐵}
14		fvres 6895	. . . . . 6 ⊢ (𝑟 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑟) = (2nd ‘𝑟))
15	14	eleq1d 2819	. . . . 5 ⊢ (𝑟 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑟) ∈ 𝐵 ↔ (2nd ‘𝑟) ∈ 𝐵))
16	15	rabbiia 3419	. . . 4 ⊢ {𝑟 ∈ (V × V) ∣ ((2nd ↾ (V × V))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵}
17	13, 16	eqtri 2758	. . 3 ⊢ (◡(2nd ↾ (V × V)) “ 𝐵) = {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵}
18	9, 17	ineq12i 4193	. 2 ⊢ ((◡(1st ↾ (V × V)) “ 𝐴) ∩ (◡(2nd ↾ (V × V)) “ 𝐵)) = ({𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵})
19		xp2 8025	. 2 ⊢ (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)}
20	1, 18, 19	3eqtr4ri 2769	1 ⊢ (𝐴 × 𝐵) = ((◡(1st ↾ (V × V)) “ 𝐴) ∩ (◡(2nd ↾ (V × V)) “ 𝐵))

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {crab 3415  Vcvv 3459   ∩ cin 3925   × cxp 5652  ^◡ccnv 5653   ↾ cres 5656   “ cima 5657   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  1^st c1st 7986  2^nd c2nd 7987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-1st 7988  df-2nd 7989

This theorem is referenced by: (None)