1stpreima - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > 1stpreima		Structured version Visualization version GIF version

Theorem 1stpreima 32796

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 7978	. . . . . 6 ⊢ (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐶)))
2	1	anbi2i 624	. . . . 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 3929	. . . . . . . 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 632	. . . . . 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 646	. . . . . . . 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 624	. . . . . . 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 646	. . . 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 6196	. . . . 5 ⊢ (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) = ((^◡1^st “ 𝐴) ∩ (𝐵 × 𝐶))
16	15	eleq2i 2829	. . . 4 ⊢ (𝑤 ∈ (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((^◡1^st “ 𝐴) ∩ (𝐵 × 𝐶)))
17		elin 3919	. . . 4 ⊢ (𝑤 ∈ ((^◡1^st “ 𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (^◡1^st “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
18		vex 3446	. . . . . 6 ⊢ 𝑤 ∈ V
19		fo1st 7963	. . . . . . 7 ⊢ 1^st :V–onto→V
20		fofn 6756	. . . . . . 7 ⊢ (1^st :V–onto→V → 1^st Fn V)
21		elpreima 7012	. . . . . . 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 710	. . . . 5 ⊢ (𝑤 ∈ (^◡1^st “ 𝐴) ↔ (1^st ‘𝑤) ∈ 𝐴)
24	23	anbi1i 625	. . . 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 7978	. . 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 2735	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 1542 ∈ wcel 2114 Vcvv 3442 ∩ cin 3902 ⊆ wss 3903 × cxp 5630 ^◡ccnv 5631 ↾ cres 5634 “ cima 5635 Fn wfn 6495 –onto→wfo 6498 ‘cfv 6500 1^st c1st 7941 2^nd c2nd 7942

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-fo 6506 df-fv 6508 df-1st 7943 df-2nd 7944

This theorem is referenced by: sxbrsigalem2 34463

W3C validator