Theorem xpinpreima 34084

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 4270	. 2 ⊢ ({𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵}) = {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)}
2		f1stres 7967	. . . . 5 ⊢ (1st ↾ (V × V)):(V × V)⟶V
3		ffn 6670	. . . . 5 ⊢ ((1st ↾ (V × V)):(V × V)⟶V → (1st ↾ (V × V)) Fn (V × V))
4		fncnvima2 7015	. . . . 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 6861	. . . . . 6 ⊢ (𝑟 ∈ (V × V) → ((1st ↾ (V × V))‘𝑟) = (1st ‘𝑟))
7	6	eleq1d 2822	. . . . 5 ⊢ (𝑟 ∈ (V × V) → (((1st ↾ (V × V))‘𝑟) ∈ 𝐴 ↔ (1st ‘𝑟) ∈ 𝐴))
8	7	rabbiia 3405	. . . 4 ⊢ {𝑟 ∈ (V × V) ∣ ((1st ↾ (V × V))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴}
9	5, 8	eqtri 2760	. . 3 ⊢ (◡(1st ↾ (V × V)) “ 𝐴) = {𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴}
10		f2ndres 7968	. . . . 5 ⊢ (2nd ↾ (V × V)):(V × V)⟶V
11		ffn 6670	. . . . 5 ⊢ ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V))
12		fncnvima2 7015	. . . . 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 6861	. . . . . 6 ⊢ (𝑟 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑟) = (2nd ‘𝑟))
15	14	eleq1d 2822	. . . . 5 ⊢ (𝑟 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑟) ∈ 𝐵 ↔ (2nd ‘𝑟) ∈ 𝐵))
16	15	rabbiia 3405	. . . 4 ⊢ {𝑟 ∈ (V × V) ∣ ((2nd ↾ (V × V))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵}
17	13, 16	eqtri 2760	. . 3 ⊢ (◡(2nd ↾ (V × V)) “ 𝐵) = {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵}
18	9, 17	ineq12i 4172	. 2 ⊢ ((◡(1st ↾ (V × V)) “ 𝐴) ∩ (◡(2nd ↾ (V × V)) “ 𝐵)) = ({𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵})
19		xp2 7980	. 2 ⊢ (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)}
20	1, 18, 19	3eqtr4ri 2771	1 ⊢ (𝐴 × 𝐵) = ((◡(1st ↾ (V × V)) “ 𝐴) ∩ (◡(2nd ↾ (V × V)) “ 𝐵))

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3401  Vcvv 3442   ∩ cin 3902   × cxp 5630  ^◡ccnv 5631   ↾ cres 5634   “ cima 5635   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  1^st c1st 7941  2^nd c2nd 7942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-1st 7943  df-2nd 7944

This theorem is referenced by: (None)