1stpreima - Mathbox for Thierry Arnoux

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Theorem 1stpreima 32531

Description: The preimage by 1^st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)

**Assertion**
Ref	Expression
1stpreima	⊢ (𝐴 ⊆ 𝐵 → (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))

Proof of Theorem 1stpreima Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp7 8026	. . . . . 6 ⊢ (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐶)))
2	1	anbi2i 621	. . . . 5 ⊢ (((1^st ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐶))))
3		anass 467	. . . . . . 7 ⊢ ((((1^st ‘𝑤) ∈ 𝐴 ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ (𝑤 ∈ (V × V) ∧ (2^nd ‘𝑤) ∈ 𝐶)) ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2^nd ‘𝑤) ∈ 𝐶))))
4	3	a1i 11	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ((((1^st ‘𝑤) ∈ 𝐴 ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ (𝑤 ∈ (V × V) ∧ (2^nd ‘𝑤) ∈ 𝐶)) ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2^nd ‘𝑤) ∈ 𝐶)))))
5		ssel 3965	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → ((1^st ‘𝑤) ∈ 𝐴 → (1^st ‘𝑤) ∈ 𝐵))
6	5	pm4.71d 560	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → ((1^st ‘𝑤) ∈ 𝐴 ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ (1^st ‘𝑤) ∈ 𝐵)))
7	6	anbi1d 629	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (((1^st ‘𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2^nd ‘𝑤) ∈ 𝐶)) ↔ (((1^st ‘𝑤) ∈ 𝐴 ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ (𝑤 ∈ (V × V) ∧ (2^nd ‘𝑤) ∈ 𝐶))))
8		an12 643	. . . . . . . 8 ⊢ ((𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐶)) ↔ ((1^st ‘𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2^nd ‘𝑤) ∈ 𝐶)))
9	8	anbi2i 621	. . . . . . 7 ⊢ (((1^st ‘𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐶))) ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2^nd ‘𝑤) ∈ 𝐶))))
10	9	a1i 11	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (((1^st ‘𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐶))) ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2^nd ‘𝑤) ∈ 𝐶)))))
11	4, 7, 10	3bitr4d 310	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (((1^st ‘𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2^nd ‘𝑤) ∈ 𝐶)) ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐶)))))
12	2, 11	bitr4id 289	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (((1^st ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2^nd ‘𝑤) ∈ 𝐶))))
13		an12 643	. . . 4 ⊢ (((1^st ‘𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2^nd ‘𝑤) ∈ 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐴 ∧ (2^nd ‘𝑤) ∈ 𝐶)))
14	12, 13	bitrdi 286	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (((1^st ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐴 ∧ (2^nd ‘𝑤) ∈ 𝐶))))
15		cnvresima 6229	. . . . 5 ⊢ (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) = ((^◡1^st “ 𝐴) ∩ (𝐵 × 𝐶))
16	15	eleq2i 2817	. . . 4 ⊢ (𝑤 ∈ (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((^◡1^st “ 𝐴) ∩ (𝐵 × 𝐶)))
17		elin 3955	. . . 4 ⊢ (𝑤 ∈ ((^◡1^st “ 𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (^◡1^st “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
18		vex 3467	. . . . . 6 ⊢ 𝑤 ∈ V
19		fo1st 8011	. . . . . . 7 ⊢ 1^st :V–onto→V
20		fofn 6808	. . . . . . 7 ⊢ (1^st :V–onto→V → 1^st Fn V)
21		elpreima 7062	. . . . . . 7 ⊢ (1^st Fn V → (𝑤 ∈ (^◡1^st “ 𝐴) ↔ (𝑤 ∈ V ∧ (1^st ‘𝑤) ∈ 𝐴)))
22	19, 20, 21	mp2b 10	. . . . . 6 ⊢ (𝑤 ∈ (^◡1^st “ 𝐴) ↔ (𝑤 ∈ V ∧ (1^st ‘𝑤) ∈ 𝐴))
23	18, 22	mpbiran 707	. . . . 5 ⊢ (𝑤 ∈ (^◡1^st “ 𝐴) ↔ (1^st ‘𝑤) ∈ 𝐴)
24	23	anbi1i 622	. . . 4 ⊢ ((𝑤 ∈ (^◡1^st “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
25	16, 17, 24	3bitri 296	. . 3 ⊢ (𝑤 ∈ (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
26		elxp7 8026	. . 3 ⊢ (𝑤 ∈ (𝐴 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐴 ∧ (2^nd ‘𝑤) ∈ 𝐶)))
27	14, 25, 26	3bitr4g 313	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑤 ∈ (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐴 × 𝐶)))
28	27	eqrdv 2723	1 ⊢ (𝐴 ⊆ 𝐵 → (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∩ cin 3938 ⊆ wss 3939 × cxp 5670 ^◡ccnv 5671 ↾ cres 5674 “ cima 5675 Fn wfn 6538 –onto→wfo 6541 ‘cfv 6543 1^st c1st 7989 2^nd c2nd 7990

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-1st 7991 df-2nd 7992

This theorem is referenced by: sxbrsigalem2 33963

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