Theorem 2ndpreima 30942

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

Assertion

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

Proof of Theorem 2ndpreima

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp7 7839	. . . . . 6 ⊢ (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐶)))
2	1	anbi1i 623	. . . . 5 ⊢ ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐶)) ∧ (2nd ‘𝑤) ∈ 𝐴))
3		ssel 3910	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐶 → ((2nd ‘𝑤) ∈ 𝐴 → (2nd ‘𝑤) ∈ 𝐶))
4	3	pm4.71rd 562	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐶 → ((2nd ‘𝑤) ∈ 𝐴 ↔ ((2nd ‘𝑤) ∈ 𝐶 ∧ (2nd ‘𝑤) ∈ 𝐴)))
5	4	anbi2d 628	. . . . . 6 ⊢ (𝐴 ⊆ 𝐶 → (((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ (2nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ ((2nd ‘𝑤) ∈ 𝐶 ∧ (2nd ‘𝑤) ∈ 𝐴))))
6		anass 468	. . . . . . . 8 ⊢ ((((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ (2nd ‘𝑤) ∈ 𝐶) ∧ (2nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ ((2nd ‘𝑤) ∈ 𝐶 ∧ (2nd ‘𝑤) ∈ 𝐴)))
7	6	bicomi 223	. . . . . . 7 ⊢ (((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ ((2nd ‘𝑤) ∈ 𝐶 ∧ (2nd ‘𝑤) ∈ 𝐴)) ↔ (((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ (2nd ‘𝑤) ∈ 𝐶) ∧ (2nd ‘𝑤) ∈ 𝐴))
8	7	a1i 11	. . . . . 6 ⊢ (𝐴 ⊆ 𝐶 → (((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ ((2nd ‘𝑤) ∈ 𝐶 ∧ (2nd ‘𝑤) ∈ 𝐴)) ↔ (((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ (2nd ‘𝑤) ∈ 𝐶) ∧ (2nd ‘𝑤) ∈ 𝐴)))
9		anass 468	. . . . . . . 8 ⊢ (((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ (2nd ‘𝑤) ∈ 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐶)))
10	9	anbi1i 623	. . . . . . 7 ⊢ ((((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ (2nd ‘𝑤) ∈ 𝐶) ∧ (2nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐶)) ∧ (2nd ‘𝑤) ∈ 𝐴))
11	10	a1i 11	. . . . . 6 ⊢ (𝐴 ⊆ 𝐶 → ((((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ (2nd ‘𝑤) ∈ 𝐶) ∧ (2nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐶)) ∧ (2nd ‘𝑤) ∈ 𝐴)))
12	5, 8, 11	3bitrd 304	. . . . 5 ⊢ (𝐴 ⊆ 𝐶 → (((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ (2nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐶)) ∧ (2nd ‘𝑤) ∈ 𝐴)))
13	2, 12	bitr4id 289	. . . 4 ⊢ (𝐴 ⊆ 𝐶 → ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ (2nd ‘𝑤) ∈ 𝐴)))
14		ancom 460	. . . 4 ⊢ ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd ‘𝑤) ∈ 𝐴) ↔ ((2nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
15		anass 468	. . . 4 ⊢ (((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ (2nd ‘𝑤) ∈ 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐴)))
16	13, 14, 15	3bitr3g 312	. . 3 ⊢ (𝐴 ⊆ 𝐶 → (((2nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐴))))
17		cnvresima 6122	. . . . 5 ⊢ (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = ((◡2nd “ 𝐴) ∩ (𝐵 × 𝐶))
18	17	eleq2i 2830	. . . 4 ⊢ (𝑤 ∈ (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((◡2nd “ 𝐴) ∩ (𝐵 × 𝐶)))
19		elin 3899	. . . 4 ⊢ (𝑤 ∈ ((◡2nd “ 𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (◡2nd “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
20		vex 3426	. . . . . 6 ⊢ 𝑤 ∈ V
21		fo2nd 7825	. . . . . . 7 ⊢ 2nd :V–onto→V
22		fofn 6674	. . . . . . 7 ⊢ (2nd :V–onto→V → 2nd Fn V)
23		elpreima 6917	. . . . . . 7 ⊢ (2nd Fn V → (𝑤 ∈ (◡2nd “ 𝐴) ↔ (𝑤 ∈ V ∧ (2nd ‘𝑤) ∈ 𝐴)))
24	21, 22, 23	mp2b 10	. . . . . 6 ⊢ (𝑤 ∈ (◡2nd “ 𝐴) ↔ (𝑤 ∈ V ∧ (2nd ‘𝑤) ∈ 𝐴))
25	20, 24	mpbiran 705	. . . . 5 ⊢ (𝑤 ∈ (◡2nd “ 𝐴) ↔ (2nd ‘𝑤) ∈ 𝐴)
26	25	anbi1i 623	. . . 4 ⊢ ((𝑤 ∈ (◡2nd “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((2nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
27	18, 19, 26	3bitri 296	. . 3 ⊢ (𝑤 ∈ (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((2nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
28		elxp7 7839	. . 3 ⊢ (𝑤 ∈ (𝐵 × 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐴)))
29	16, 27, 28	3bitr4g 313	. 2 ⊢ (𝐴 ⊆ 𝐶 → (𝑤 ∈ (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐵 × 𝐴)))
30	29	eqrdv 2736	1 ⊢ (𝐴 ⊆ 𝐶 → (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883   × cxp 5578  ^◡ccnv 5579   ↾ cres 5582   “ cima 5583   Fn wfn 6413  –onto→wfo 6416  ‘cfv 6418  1^st c1st 7802  2^nd c2nd 7803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426  df-1st 7804  df-2nd 7805

This theorem is referenced by:  sxbrsigalem2  32153