xpinpreima2 - Mathbox for Thierry Arnoux

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Theorem xpinpreima2 34589

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 8042	. . . 4 ⊢ (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵)}
2		xpss 5667	. . . . . 6 ⊢ (𝐸 × 𝐹) ⊆ (V × V)
3		rabss2 4001	. . . . . 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 14	. . . . 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 783	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → 𝑟 ∈ (V × V))
6		simpll 779	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → 𝐴 ⊆ 𝐸)
7		simprrl 793	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → (1^st ‘𝑟) ∈ 𝐴)
8	6, 7	sseldd 3910	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → (1^st ‘𝑟) ∈ 𝐸)
9		simplr 781	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → 𝐵 ⊆ 𝐹)
10		simprrr 794	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → (2^nd ‘𝑟) ∈ 𝐵)
11	9, 10	sseldd 3910	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → (2^nd ‘𝑟) ∈ 𝐹)
12	8, 11	jca 521	. . . . . . 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 8040	. . . . . . 7 ⊢ (𝑟 ∈ (𝐸 × 𝐹) ↔ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐸 ∧ (2^nd ‘𝑟) ∈ 𝐹)))
14	5, 12, 13	sylanbrc 595	. . . . . 6 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵))) → 𝑟 ∈ (𝐸 × 𝐹))
15	14	rabss3d 4005	. . . . 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 3926	. . . 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 2814	. . 3 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1^st ‘𝑟) ∈ 𝐴 ∧ (2^nd ‘𝑟) ∈ 𝐵)})
18		inrab 4260	. . 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 2813	. 2 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1^st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2^nd ‘𝑟) ∈ 𝐵}))
20		f1stres 8029	. . . . 5 ⊢ (1^st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸
21		ffn 6714	. . . . 5 ⊢ ((1^st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸 → (1^st ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
22		fncnvima2 7065	. . . . 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 6909	. . . . . 6 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → ((1^st ↾ (𝐸 × 𝐹))‘𝑟) = (1^st ‘𝑟))
25	24	eleq1d 2845	. . . . 5 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → (((1^st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴 ↔ (1^st ‘𝑟) ∈ 𝐴))
26	25	rabbiia 3416	. . . 4 ⊢ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1^st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1^st ‘𝑟) ∈ 𝐴}
27	23, 26	eqtri 2783	. . 3 ⊢ (^◡(1^st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1^st ‘𝑟) ∈ 𝐴}
28		f2ndres 8030	. . . . 5 ⊢ (2^nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹
29		ffn 6714	. . . . 5 ⊢ ((2^nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹 → (2^nd ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
30		fncnvima2 7065	. . . . 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 6909	. . . . . 6 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → ((2^nd ↾ (𝐸 × 𝐹))‘𝑟) = (2^nd ‘𝑟))
33	32	eleq1d 2845	. . . . 5 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → (((2^nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵 ↔ (2^nd ‘𝑟) ∈ 𝐵))
34	33	rabbiia 3416	. . . 4 ⊢ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((2^nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2^nd ‘𝑟) ∈ 𝐵}
35	31, 34	eqtri 2783	. . 3 ⊢ (^◡(2^nd ↾ (𝐸 × 𝐹)) “ 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2^nd ‘𝑟) ∈ 𝐵}
36	27, 35	ineq12i 4114	. 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 2813	1 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ((^◡(1^st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ (^◡(2^nd ↾ (𝐸 × 𝐹)) “ 𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 {crab 3412 Vcvv 3450 ⊆ wss 3896 ∩ cin 4088 × cxp 5649 ^◡ccnv 5650 ↾ cres 5653 “ cima 5654 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 1^st c1st 8003 2^nd c2nd 8004

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5391 ax-un 7755

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-1st 8005 df-2nd 8006

This theorem is used by: cnre2csqima 34593 sxbrsigalem2 34968 sxbrsiga 34972

W3C validator