xpinpreima2 - Mathbox for Thierry Arnoux

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Theorem xpinpreima2 33888

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
xpinpreima2	⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ((^◡(1^st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ (^◡(2^nd ↾ (𝐸 × 𝐹)) “ 𝐵)))

Proof of Theorem xpinpreima2 Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xp2 7952	. . . 4 ⊢ (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵)}
2		xpss 5629	. . . . . 6 ⊢ (𝐸 × 𝐹) ⊆ (V × V)
3		rabss2 4025	. . . . . 6 ⊢ ((𝐸 × 𝐹) ⊆ (V × V) → {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵)} ⊆ {𝑟 ∈ (V × V) ∣ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵)})
4	2, 3	mp1i 13	. . . . 5 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵)} ⊆ {𝑟 ∈ (V × V) ∣ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵)})
5		simprl 770	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → 𝑟 ∈ (V × V))
6		simpll 766	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → 𝐴 ⊆ 𝐸)
7		simprrl 780	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → (1^st ‘𝑟) ∈ 𝐴)
8	6, 7	sseldd 3932	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → (1^st ‘𝑟) ∈ 𝐸)
9		simplr 768	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → 𝐵 ⊆ 𝐹)
10		simprrr 781	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → (2^nd ‘𝑟) ∈ 𝐵)
11	9, 10	sseldd 3932	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → (2^nd ‘𝑟) ∈ 𝐹)
12	8, 11	jca 511	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → ((1^st ‘𝑟) ∈ 𝐸 ∧ (2^nd ‘𝑟) ∈ 𝐹))
13		elxp7 7950	. . . . . . 7 ⊢ (𝑟 ∈ (𝐸 × 𝐹) ↔ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐸 ∧ (2^nd ‘𝑟) ∈ 𝐹)))
14	5, 12, 13	sylanbrc 583	. . . . . 6 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → 𝑟 ∈ (𝐸 × 𝐹))
15	14	rabss3d 4028	. . . . 5 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → {𝑟 ∈ (V × V) ∣ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵)} ⊆ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵)})
16	4, 15	eqssd 3949	. . . 4 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵)} = {𝑟 ∈ (V × V) ∣ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵)})
17	1, 16	eqtr4id 2783	. . 3 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵)})
18		inrab 4263	. . 3 ⊢ ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1^st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2^nd ‘𝑟) ∈ 𝐵}) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵)}
19	17, 18	eqtr4di 2782	. 2 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1^st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2^nd ‘𝑟) ∈ 𝐵}))
20		f1stres 7939	. . . . 5 ⊢ (1^st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸
21		ffn 6646	. . . . 5 ⊢ ((1^st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸 → (1^st ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
22		fncnvima2 6988	. . . . 5 ⊢ ((1^st ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹) → (^◡(1^st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1^st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴})
23	20, 21, 22	mp2b 10	. . . 4 ⊢ (^◡(1^st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1^st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴}
24		fvres 6835	. . . . . 6 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → ((1^st ↾ (𝐸 × 𝐹))‘𝑟) = (1^st ‘𝑟))
25	24	eleq1d 2813	. . . . 5 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → (((1^st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴 ↔ (1^st ‘𝑟) ∈ 𝐴))
26	25	rabbiia 3396	. . . 4 ⊢ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1^st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1^st ‘𝑟) ∈ 𝐴}
27	23, 26	eqtri 2752	. . 3 ⊢ (^◡(1^st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1^st ‘𝑟) ∈ 𝐴}
28		f2ndres 7940	. . . . 5 ⊢ (2^nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹
29		ffn 6646	. . . . 5 ⊢ ((2^nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹 → (2^nd ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
30		fncnvima2 6988	. . . . 5 ⊢ ((2^nd ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹) → (^◡(2^nd ↾ (𝐸 × 𝐹)) “ 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((2^nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵})
31	28, 29, 30	mp2b 10	. . . 4 ⊢ (^◡(2^nd ↾ (𝐸 × 𝐹)) “ 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((2^nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵}
32		fvres 6835	. . . . . 6 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → ((2^nd ↾ (𝐸 × 𝐹))‘𝑟) = (2^nd ‘𝑟))
33	32	eleq1d 2813	. . . . 5 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → (((2^nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵 ↔ (2^nd ‘𝑟) ∈ 𝐵))
34	33	rabbiia 3396	. . . 4 ⊢ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((2^nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2^nd ‘𝑟) ∈ 𝐵}
35	31, 34	eqtri 2752	. . 3 ⊢ (^◡(2^nd ↾ (𝐸 × 𝐹)) “ 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2^nd ‘𝑟) ∈ 𝐵}
36	27, 35	ineq12i 4165	. 2 ⊢ ((^◡(1^st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ (^◡(2^nd ↾ (𝐸 × 𝐹)) “ 𝐵)) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1^st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2^nd ‘𝑟) ∈ 𝐵})
37	19, 36	eqtr4di 2782	1 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ((^◡(1^st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ (^◡(2^nd ↾ (𝐸 × 𝐹)) “ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3392 Vcvv 3433 ∩ cin 3898 ⊆ wss 3899 × cxp 5611 ^◡ccnv 5612 ↾ cres 5615 “ cima 5616 Fn wfn 6471 ⟶wf 6472 ‘cfv 6476 1^st c1st 7913 2^nd c2nd 7914

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5231 ax-nul 5241 ax-pr 5367 ax-un 7662

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3393 df-v 3435 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4940 df-br 5089 df-opab 5151 df-mpt 5170 df-id 5508 df-xp 5619 df-rel 5620 df-cnv 5621 df-co 5622 df-dm 5623 df-rn 5624 df-res 5625 df-ima 5626 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-fv 6484 df-1st 7915 df-2nd 7916

This theorem is referenced by: cnre2csqima 33892 sxbrsigalem2 34267 sxbrsiga 34271

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