2ndpreima - Mathbox for Thierry Arnoux

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Theorem 2ndpreima 31929

Description: The preimage by 2^nd is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)

**Assertion**
Ref	Expression
2ndpreima	⊢ (𝐴 ⊆ 𝐶 → (^◡(2^nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))

Proof of Theorem 2ndpreima Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp7 8010	. . . . . 6 ⊢ (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐶)))
2	1	anbi1i 625	. . . . 5 ⊢ ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2^nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐶)) ∧ (2^nd ‘𝑤) ∈ 𝐴))
3		ssel 3976	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐶 → ((2^nd ‘𝑤) ∈ 𝐴 → (2^nd ‘𝑤) ∈ 𝐶))
4	3	pm4.71rd 564	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐶 → ((2^nd ‘𝑤) ∈ 𝐴 ↔ ((2^nd ‘𝑤) ∈ 𝐶 ∧ (2^nd ‘𝑤) ∈ 𝐴)))
5	4	anbi2d 630	. . . . . 6 ⊢ (𝐴 ⊆ 𝐶 → (((𝑤 ∈ (V × V) ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ (2^nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ ((2^nd ‘𝑤) ∈ 𝐶 ∧ (2^nd ‘𝑤) ∈ 𝐴))))
6		anass 470	. . . . . . . 8 ⊢ ((((𝑤 ∈ (V × V) ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ (2^nd ‘𝑤) ∈ 𝐶) ∧ (2^nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ ((2^nd ‘𝑤) ∈ 𝐶 ∧ (2^nd ‘𝑤) ∈ 𝐴)))
7	6	bicomi 223	. . . . . . 7 ⊢ (((𝑤 ∈ (V × V) ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ ((2^nd ‘𝑤) ∈ 𝐶 ∧ (2^nd ‘𝑤) ∈ 𝐴)) ↔ (((𝑤 ∈ (V × V) ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ (2^nd ‘𝑤) ∈ 𝐶) ∧ (2^nd ‘𝑤) ∈ 𝐴))
8	7	a1i 11	. . . . . 6 ⊢ (𝐴 ⊆ 𝐶 → (((𝑤 ∈ (V × V) ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ ((2^nd ‘𝑤) ∈ 𝐶 ∧ (2^nd ‘𝑤) ∈ 𝐴)) ↔ (((𝑤 ∈ (V × V) ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ (2^nd ‘𝑤) ∈ 𝐶) ∧ (2^nd ‘𝑤) ∈ 𝐴)))
9		anass 470	. . . . . . . 8 ⊢ (((𝑤 ∈ (V × V) ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ (2^nd ‘𝑤) ∈ 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐶)))
10	9	anbi1i 625	. . . . . . 7 ⊢ ((((𝑤 ∈ (V × V) ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ (2^nd ‘𝑤) ∈ 𝐶) ∧ (2^nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐶)) ∧ (2^nd ‘𝑤) ∈ 𝐴))
11	10	a1i 11	. . . . . 6 ⊢ (𝐴 ⊆ 𝐶 → ((((𝑤 ∈ (V × V) ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ (2^nd ‘𝑤) ∈ 𝐶) ∧ (2^nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐶)) ∧ (2^nd ‘𝑤) ∈ 𝐴)))
12	5, 8, 11	3bitrd 305	. . . . 5 ⊢ (𝐴 ⊆ 𝐶 → (((𝑤 ∈ (V × V) ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ (2^nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐶)) ∧ (2^nd ‘𝑤) ∈ 𝐴)))
13	2, 12	bitr4id 290	. . . 4 ⊢ (𝐴 ⊆ 𝐶 → ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2^nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ (2^nd ‘𝑤) ∈ 𝐴)))
14		ancom 462	. . . 4 ⊢ ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2^nd ‘𝑤) ∈ 𝐴) ↔ ((2^nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
15		anass 470	. . . 4 ⊢ (((𝑤 ∈ (V × V) ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ (2^nd ‘𝑤) ∈ 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐴)))
16	13, 14, 15	3bitr3g 313	. . 3 ⊢ (𝐴 ⊆ 𝐶 → (((2^nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐴))))
17		cnvresima 6230	. . . . 5 ⊢ (^◡(2^nd ↾ (𝐵 × 𝐶)) “ 𝐴) = ((^◡2^nd “ 𝐴) ∩ (𝐵 × 𝐶))
18	17	eleq2i 2826	. . . 4 ⊢ (𝑤 ∈ (^◡(2^nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((^◡2^nd “ 𝐴) ∩ (𝐵 × 𝐶)))
19		elin 3965	. . . 4 ⊢ (𝑤 ∈ ((^◡2^nd “ 𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (^◡2^nd “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
20		vex 3479	. . . . . 6 ⊢ 𝑤 ∈ V
21		fo2nd 7996	. . . . . . 7 ⊢ 2^nd :V–onto→V
22		fofn 6808	. . . . . . 7 ⊢ (2^nd :V–onto→V → 2^nd Fn V)
23		elpreima 7060	. . . . . . 7 ⊢ (2^nd Fn V → (𝑤 ∈ (^◡2^nd “ 𝐴) ↔ (𝑤 ∈ V ∧ (2^nd ‘𝑤) ∈ 𝐴)))
24	21, 22, 23	mp2b 10	. . . . . 6 ⊢ (𝑤 ∈ (^◡2^nd “ 𝐴) ↔ (𝑤 ∈ V ∧ (2^nd ‘𝑤) ∈ 𝐴))
25	20, 24	mpbiran 708	. . . . 5 ⊢ (𝑤 ∈ (^◡2^nd “ 𝐴) ↔ (2^nd ‘𝑤) ∈ 𝐴)
26	25	anbi1i 625	. . . 4 ⊢ ((𝑤 ∈ (^◡2^nd “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((2^nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
27	18, 19, 26	3bitri 297	. . 3 ⊢ (𝑤 ∈ (^◡(2^nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((2^nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
28		elxp7 8010	. . 3 ⊢ (𝑤 ∈ (𝐵 × 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐴)))
29	16, 27, 28	3bitr4g 314	. 2 ⊢ (𝐴 ⊆ 𝐶 → (𝑤 ∈ (^◡(2^nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐵 × 𝐴)))
30	29	eqrdv 2731	1 ⊢ (𝐴 ⊆ 𝐶 → (^◡(2^nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∩ cin 3948 ⊆ wss 3949 × cxp 5675 ^◡ccnv 5676 ↾ cres 5679 “ cima 5680 Fn wfn 6539 –onto→wfo 6542 ‘cfv 6544 1^st c1st 7973 2^nd c2nd 7974

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fo 6550 df-fv 6552 df-1st 7975 df-2nd 7976

This theorem is referenced by: sxbrsigalem2 33285

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