1stpreima - Mathbox for Thierry Arnoux

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Theorem 1stpreima 32780

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 7977	. . . . . 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 3915	. . . . . . . 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 6194	. . . . 5 ⊢ (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) = ((^◡1^st “ 𝐴) ∩ (𝐵 × 𝐶))
16	15	eleq2i 2828	. . . 4 ⊢ (𝑤 ∈ (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((^◡1^st “ 𝐴) ∩ (𝐵 × 𝐶)))
17		elin 3905	. . . 4 ⊢ (𝑤 ∈ ((^◡1^st “ 𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (^◡1^st “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
18		vex 3433	. . . . . 6 ⊢ 𝑤 ∈ V
19		fo1st 7962	. . . . . . 7 ⊢ 1^st :V–onto→V
20		fofn 6754	. . . . . . 7 ⊢ (1^st :V–onto→V → 1^st Fn V)
21		elpreima 7010	. . . . . . 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 7977	. . 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 2734	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 3429 ∩ cin 3888 ⊆ wss 3889 × cxp 5629 ^◡ccnv 5630 ↾ cres 5633 “ cima 5634 Fn wfn 6493 –onto→wfo 6496 ‘cfv 6498 1^st c1st 7940 2^nd c2nd 7941

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 2708 ax-sep 5231 ax-nul 5241 ax-pr 5375 ax-un 7689

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-fo 6504 df-fv 6506 df-1st 7942 df-2nd 7943

This theorem is referenced by: sxbrsigalem2 34430

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