Theorem xpinpreima 34296

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 4269	. 2 ⊢ ({𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵}) = {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)}
2		f1stres 8006	. . . . 5 ⊢ (1st ↾ (V × V)):(V × V)⟶V
3		ffn 6705	. . . . 5 ⊢ ((1st ↾ (V × V)):(V × V)⟶V → (1st ↾ (V × V)) Fn (V × V))
4		fncnvima2 7056	. . . . 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 6900	. . . . . 6 ⊢ (𝑟 ∈ (V × V) → ((1st ↾ (V × V))‘𝑟) = (1st ‘𝑟))
7	6	eleq1d 2848	. . . . 5 ⊢ (𝑟 ∈ (V × V) → (((1st ↾ (V × V))‘𝑟) ∈ 𝐴 ↔ (1st ‘𝑟) ∈ 𝐴))
8	7	rabbiia 3420	. . . 4 ⊢ {𝑟 ∈ (V × V) ∣ ((1st ↾ (V × V))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴}
9	5, 8	eqtri 2786	. . 3 ⊢ (◡(1st ↾ (V × V)) “ 𝐴) = {𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴}
10		f2ndres 8007	. . . . 5 ⊢ (2nd ↾ (V × V)):(V × V)⟶V
11		ffn 6705	. . . . 5 ⊢ ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V))
12		fncnvima2 7056	. . . . 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 6900	. . . . . 6 ⊢ (𝑟 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑟) = (2nd ‘𝑟))
15	14	eleq1d 2848	. . . . 5 ⊢ (𝑟 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑟) ∈ 𝐵 ↔ (2nd ‘𝑟) ∈ 𝐵))
16	15	rabbiia 3420	. . . 4 ⊢ {𝑟 ∈ (V × V) ∣ ((2nd ↾ (V × V))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵}
17	13, 16	eqtri 2786	. . 3 ⊢ (◡(2nd ↾ (V × V)) “ 𝐵) = {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵}
18	9, 17	ineq12i 4171	. 2 ⊢ ((◡(1st ↾ (V × V)) “ 𝐴) ∩ (◡(2nd ↾ (V × V)) “ 𝐵)) = ({𝑟 ∈ (V × V) ∣ (1st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (V × V) ∣ (2nd ‘𝑟) ∈ 𝐵})
19		xp2 8019	. 2 ⊢ (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)}
20	1, 18, 19	3eqtr4ri 2797	1 ⊢ (𝐴 × 𝐵) = ((◡(1st ↾ (V × V)) “ 𝐴) ∩ (◡(2nd ↾ (V × V)) “ 𝐵))

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455   ∩ cin 3904   × cxp 5659  ^◡ccnv 5660   ↾ cres 5663   “ cima 5664   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  1^st c1st 7980  2^nd c2nd 7981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-1st 7982  df-2nd 7983

This theorem is referenced by: (None)