1stpreima - Mathbox for Thierry Arnoux

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Theorem 1stpreima 33081

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 635	. . . . 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 474	. . . . . . 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 3932	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → ((1^st ‘𝑤) ∈ 𝐴 → (1^st ‘𝑤) ∈ 𝐵))
6	5	pm4.71d 571	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → ((1^st ‘𝑤) ∈ 𝐴 ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ (1^st ‘𝑤) ∈ 𝐵)))
7	6	anbi1d 643	. . . . . 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 658	. . . . . . . 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 635	. . . . . . 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 314	. . . . 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 293	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (((1^st ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2^nd ‘𝑤) ∈ 𝐶))))
13		an12 658	. . . 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 290	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (((1^st ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐴 ∧ (2^nd ‘𝑤) ∈ 𝐶))))
15		cnvresima 6233	. . . . 5 ⊢ (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) = ((^◡1^st “ 𝐴) ∩ (𝐵 × 𝐶))
16	15	eleq2i 2857	. . . 4 ⊢ (𝑤 ∈ (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((^◡1^st “ 𝐴) ∩ (𝐵 × 𝐶)))
17		elin 3922	. . . 4 ⊢ (𝑤 ∈ ((^◡1^st “ 𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (^◡1^st “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
18		vex 3461	. . . . . 6 ⊢ 𝑤 ∈ V
19		fo1st 8008	. . . . . . 7 ⊢ 1^st :V–onto→V
20		fofn 6798	. . . . . . 7 ⊢ (1^st :V–onto→V → 1^st Fn V)
21		elpreima 7057	. . . . . . 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 722	. . . . 5 ⊢ (𝑤 ∈ (^◡1^st “ 𝐴) ↔ (1^st ‘𝑤) ∈ 𝐴)
24	23	anbi1i 636	. . . 4 ⊢ ((𝑤 ∈ (^◡1^st “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1^st ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
25	16, 17, 24	3bitri 300	. . 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 317	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑤 ∈ (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐴 × 𝐶)))
28	27	eqrdv 2763	1 ⊢ (𝐴 ⊆ 𝐵 → (^◡(1^st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 Vcvv 3457 ∩ cin 3905 ⊆ wss 3906 × cxp 5661 ^◡ccnv 5662 ↾ cres 5665 “ cima 5666 Fn wfn 6535 –onto→wfo 6538 ‘cfv 6540 1^st c1st 7986 2^nd c2nd 7987

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pr 5406 ax-un 7738

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-fo 6546 df-fv 6548 df-1st 7988 df-2nd 7989

This theorem is used by: sxbrsigalem2 34700

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