2ndpreima - Mathbox for Thierry Arnoux

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Theorem 2ndpreima 31967

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 8012	. . . . . 6 ⊢ (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐶)))
2	1	anbi1i 624	. . . . 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 3975	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐶 → ((2^nd ‘𝑤) ∈ 𝐴 → (2^nd ‘𝑤) ∈ 𝐶))
4	3	pm4.71rd 563	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐶 → ((2^nd ‘𝑤) ∈ 𝐴 ↔ ((2^nd ‘𝑤) ∈ 𝐶 ∧ (2^nd ‘𝑤) ∈ 𝐴)))
5	4	anbi2d 629	. . . . . 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 469	. . . . . . . 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 469	. . . . . . . 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 624	. . . . . . 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 304	. . . . 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 289	. . . 4 ⊢ (𝐴 ⊆ 𝐶 → ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2^nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1^st ‘𝑤) ∈ 𝐵) ∧ (2^nd ‘𝑤) ∈ 𝐴)))
14		ancom 461	. . . 4 ⊢ ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2^nd ‘𝑤) ∈ 𝐴) ↔ ((2^nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
15		anass 469	. . . 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 312	. . 3 ⊢ (𝐴 ⊆ 𝐶 → (((2^nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐴))))
17		cnvresima 6229	. . . . 5 ⊢ (^◡(2^nd ↾ (𝐵 × 𝐶)) “ 𝐴) = ((^◡2^nd “ 𝐴) ∩ (𝐵 × 𝐶))
18	17	eleq2i 2825	. . . 4 ⊢ (𝑤 ∈ (^◡(2^nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((^◡2^nd “ 𝐴) ∩ (𝐵 × 𝐶)))
19		elin 3964	. . . 4 ⊢ (𝑤 ∈ ((^◡2^nd “ 𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (^◡2^nd “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
20		vex 3478	. . . . . 6 ⊢ 𝑤 ∈ V
21		fo2nd 7998	. . . . . . 7 ⊢ 2^nd :V–onto→V
22		fofn 6807	. . . . . . 7 ⊢ (2^nd :V–onto→V → 2^nd Fn V)
23		elpreima 7059	. . . . . . 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 707	. . . . 5 ⊢ (𝑤 ∈ (^◡2^nd “ 𝐴) ↔ (2^nd ‘𝑤) ∈ 𝐴)
26	25	anbi1i 624	. . . 4 ⊢ ((𝑤 ∈ (^◡2^nd “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((2^nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
27	18, 19, 26	3bitri 296	. . 3 ⊢ (𝑤 ∈ (^◡(2^nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((2^nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
28		elxp7 8012	. . 3 ⊢ (𝑤 ∈ (𝐵 × 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) ∈ 𝐵 ∧ (2^nd ‘𝑤) ∈ 𝐴)))
29	16, 27, 28	3bitr4g 313	. 2 ⊢ (𝐴 ⊆ 𝐶 → (𝑤 ∈ (^◡(2^nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐵 × 𝐴)))
30	29	eqrdv 2730	1 ⊢ (𝐴 ⊆ 𝐶 → (^◡(2^nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∩ cin 3947 ⊆ wss 3948 × cxp 5674 ^◡ccnv 5675 ↾ cres 5678 “ cima 5679 Fn wfn 6538 –onto→wfo 6541 ‘cfv 6543 1^st c1st 7975 2^nd c2nd 7976

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-1st 7977 df-2nd 7978

This theorem is referenced by: sxbrsigalem2 33354

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