Theorem xpinpreima 33991

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 4265	. 2 ⊢ ({𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵}) = {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)}
2		f1stres 7954	. . . . 5 ⊢ (1st ↾ (V × V)):(V × V)⟶V
3		ffn 6659	. . . . 5 ⊢ ((1st ↾ (V × V)):(V × V)⟶V → (1st ↾ (V × V)) Fn (V × V))
4		fncnvima2 7003	. . . . 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 6850	. . . . . 6 ⊢ (𝑟 ∈ (V × V) → ((1st ↾ (V × V))‘𝑟) = (1st ‘𝑟))
7	6	eleq1d 2818	. . . . 5 ⊢ (𝑟 ∈ (V × V) → (((1st ↾ (V × V))‘𝑟) ∈ 𝐴 ↔ (1st ‘𝑟) ∈ 𝐴))
8	7	rabbiia 3400	. . . 4 ⊢ {𝑟 ∈ (V × V) ∣ ((1st ↾ (V × V))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴}
9	5, 8	eqtri 2756	. . 3 ⊢ (◡(1st ↾ (V × V)) “ 𝐴) = {𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴}
10		f2ndres 7955	. . . . 5 ⊢ (2nd ↾ (V × V)):(V × V)⟶V
11		ffn 6659	. . . . 5 ⊢ ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V))
12		fncnvima2 7003	. . . . 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 6850	. . . . . 6 ⊢ (𝑟 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑟) = (2nd ‘𝑟))
15	14	eleq1d 2818	. . . . 5 ⊢ (𝑟 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑟) ∈ 𝐵 ↔ (2nd ‘𝑟) ∈ 𝐵))
16	15	rabbiia 3400	. . . 4 ⊢ {𝑟 ∈ (V × V) ∣ ((2nd ↾ (V × V))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵}
17	13, 16	eqtri 2756	. . 3 ⊢ (◡(2nd ↾ (V × V)) “ 𝐵) = {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵}
18	9, 17	ineq12i 4167	. 2 ⊢ ((◡(1st ↾ (V × V)) “ 𝐴) ∩ (◡(2nd ↾ (V × V)) “ 𝐵)) = ({𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵})
19		xp2 7967	. 2 ⊢ (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)}
20	1, 18, 19	3eqtr4ri 2767	1 ⊢ (𝐴 × 𝐵) = ((◡(1st ↾ (V × V)) “ 𝐴) ∩ (◡(2nd ↾ (V × V)) “ 𝐵))

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3396  Vcvv 3437   ∩ cin 3897   × cxp 5619  ^◡ccnv 5620   ↾ cres 5623   “ cima 5624   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489  1^st c1st 7928  2^nd c2nd 7929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-1st 7930  df-2nd 7931

This theorem is referenced by: (None)