xpinpreima2 - Mathbox for Thierry Arnoux

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Theorem xpinpreima2 33920

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 7958	. . . 4 ⊢ (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵)}
2		xpss 5630	. . . . . 6 ⊢ (𝐸 × 𝐹) ⊆ (V × V)
3		rabss2 4024	. . . . . 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 3930	. . . . . . . 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 3930	. . . . . . . 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 7956	. . . . . . 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 3947	. . . 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 2785	. . 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 2784	. 2 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1^st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2^nd ‘𝑟) ∈ 𝐵}))
20		f1stres 7945	. . . . 5 ⊢ (1^st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸
21		ffn 6651	. . . . 5 ⊢ ((1^st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸 → (1^st ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
22		fncnvima2 6994	. . . . 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 6841	. . . . . 6 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → ((1^st ↾ (𝐸 × 𝐹))‘𝑟) = (1^st ‘𝑟))
25	24	eleq1d 2816	. . . . 5 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → (((1^st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴 ↔ (1^st ‘𝑟) ∈ 𝐴))
26	25	rabbiia 3399	. . . 4 ⊢ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1^st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1^st ‘𝑟) ∈ 𝐴}
27	23, 26	eqtri 2754	. . 3 ⊢ (^◡(1^st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1^st ‘𝑟) ∈ 𝐴}
28		f2ndres 7946	. . . . 5 ⊢ (2^nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹
29		ffn 6651	. . . . 5 ⊢ ((2^nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹 → (2^nd ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
30		fncnvima2 6994	. . . . 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 6841	. . . . . 6 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → ((2^nd ↾ (𝐸 × 𝐹))‘𝑟) = (2^nd ‘𝑟))
33	32	eleq1d 2816	. . . . 5 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → (((2^nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵 ↔ (2^nd ‘𝑟) ∈ 𝐵))
34	33	rabbiia 3399	. . . 4 ⊢ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((2^nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2^nd ‘𝑟) ∈ 𝐵}
35	31, 34	eqtri 2754	. . 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 2784	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 1541 ∈ wcel 2111 {crab 3395 Vcvv 3436 ∩ cin 3896 ⊆ wss 3897 × cxp 5612 ^◡ccnv 5613 ↾ cres 5616 “ cima 5617 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 1^st c1st 7919 2^nd c2nd 7920

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7668

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-fv 6489 df-1st 7921 df-2nd 7922

This theorem is referenced by: cnre2csqima 33924 sxbrsigalem2 34299 sxbrsiga 34303

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