1stpreima - Mathbox for Thierry Arnoux

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Theorem 1stpreima 32684

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 8023	. . . . . 6 ⊢ (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐶)))
2	1	anbi2i 623	. . . . 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 468	. . . . . . 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 3952	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → ((1^st ‘𝑤) ∈ 𝐴 → (1^st ‘𝑤) ∈ 𝐵))
6	5	pm4.71d 561	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → ((1^st ‘𝑤) ∈ 𝐴 ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ (1^st ‘𝑤) ∈ 𝐵)))
7	6	anbi1d 631	. . . . . 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 645	. . . . . . . 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 623	. . . . . . 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 311	. . . . 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 290	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (((1^st ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2^nd ‘𝑤) ∈ 𝐶))))
13		an12 645	. . . 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 287	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (((1^st ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐴 ∧ (2^nd ‘𝑤) ∈ 𝐶))))
15		cnvresima 6219	. . . . 5 ⊢ (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) = ((^◡1^st “ 𝐴) ∩ (𝐵 × 𝐶))
16	15	eleq2i 2826	. . . 4 ⊢ (𝑤 ∈ (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((^◡1^st “ 𝐴) ∩ (𝐵 × 𝐶)))
17		elin 3942	. . . 4 ⊢ (𝑤 ∈ ((^◡1^st “ 𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (^◡1^st “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
18		vex 3463	. . . . . 6 ⊢ 𝑤 ∈ V
19		fo1st 8008	. . . . . . 7 ⊢ 1^st :V–onto→V
20		fofn 6792	. . . . . . 7 ⊢ (1^st :V–onto→V → 1^st Fn V)
21		elpreima 7048	. . . . . . 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 709	. . . . 5 ⊢ (𝑤 ∈ (^◡1^st “ 𝐴) ↔ (1^st ‘𝑤) ∈ 𝐴)
24	23	anbi1i 624	. . . 4 ⊢ ((𝑤 ∈ (^◡1^st “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
25	16, 17, 24	3bitri 297	. . 3 ⊢ (𝑤 ∈ (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
26		elxp7 8023	. . 3 ⊢ (𝑤 ∈ (𝐴 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐴 ∧ (2^nd ‘𝑤) ∈ 𝐶)))
27	14, 25, 26	3bitr4g 314	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑤 ∈ (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐴 × 𝐶)))
28	27	eqrdv 2733	1 ⊢ (𝐴 ⊆ 𝐵 → (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3459 ∩ cin 3925 ⊆ wss 3926 × cxp 5652 ^◡ccnv 5653 ↾ cres 5656 “ cima 5657 Fn wfn 6526 –onto→wfo 6529 ‘cfv 6531 1^st c1st 7986 2^nd c2nd 7987

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402 ax-un 7729

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-fo 6537 df-fv 6539 df-1st 7988 df-2nd 7989

This theorem is referenced by: sxbrsigalem2 34318

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