Theorem xpinpreima 31865

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 4246	. 2 ⊢ ({𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵}) = {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)}
2		f1stres 7849	. . . . 5 ⊢ (1st ↾ (V × V)):(V × V)⟶V
3		ffn 6598	. . . . 5 ⊢ ((1st ↾ (V × V)):(V × V)⟶V → (1st ↾ (V × V)) Fn (V × V))
4		fncnvima2 6935	. . . . 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 6790	. . . . . 6 ⊢ (𝑟 ∈ (V × V) → ((1st ↾ (V × V))‘𝑟) = (1st ‘𝑟))
7	6	eleq1d 2825	. . . . 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 2768	. . 3 ⊢ (◡(1st ↾ (V × V)) “ 𝐴) = {𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴}
10		f2ndres 7850	. . . . 5 ⊢ (2nd ↾ (V × V)):(V × V)⟶V
11		ffn 6598	. . . . 5 ⊢ ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V))
12		fncnvima2 6935	. . . . 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 6790	. . . . . 6 ⊢ (𝑟 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑟) = (2nd ‘𝑟))
15	14	eleq1d 2825	. . . . 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 2768	. . 3 ⊢ (◡(2nd ↾ (V × V)) “ 𝐵) = {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵}
18	9, 17	ineq12i 4150	. 2 ⊢ ((◡(1st ↾ (V × V)) “ 𝐴) ∩ (◡(2nd ↾ (V × V)) “ 𝐵)) = ({𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵})
19		xp2 7862	. 2 ⊢ (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)}
20	1, 18, 19	3eqtr4ri 2779	1 ⊢ (𝐴 × 𝐵) = ((◡(1st ↾ (V × V)) “ 𝐴) ∩ (◡(2nd ↾ (V × V)) “ 𝐵))

Syntax hints:   ∧ wa 396   = wceq 1542   ∈ wcel 2110  {crab 3070  Vcvv 3431   ∩ cin 3891   × cxp 5588  ^◡ccnv 5589   ↾ cres 5592   “ cima 5593   Fn wfn 6427  ⟶wf 6428  ‘cfv 6432  1^st c1st 7823  2^nd c2nd 7824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7583

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-fv 6440  df-1st 7825  df-2nd 7826

This theorem is referenced by: (None)