Theorem 2ndpreima 32797

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 7978	. . . . . 6 ⊢ (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐶)))
2	1	anbi1i 625	. . . . 5 ⊢ ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐶)) ∧ (2nd ‘𝑤) ∈ 𝐴))
3		ssel 3929	. . . . . . . 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 631	. . . . . 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 224	. . . . . . 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 625	. . . . . . 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 305	. . . . 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 290	. . . 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 313	. . 3 ⊢ (𝐴 ⊆ 𝐶 → (((2nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐴))))
17		cnvresima 6196	. . . . 5 ⊢ (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = ((◡2nd “ 𝐴) ∩ (𝐵 × 𝐶))
18	17	eleq2i 2829	. . . 4 ⊢ (𝑤 ∈ (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((◡2nd “ 𝐴) ∩ (𝐵 × 𝐶)))
19		elin 3919	. . . 4 ⊢ (𝑤 ∈ ((◡2nd “ 𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (◡2nd “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
20		vex 3446	. . . . . 6 ⊢ 𝑤 ∈ V
21		fo2nd 7964	. . . . . . 7 ⊢ 2nd :V–onto→V
22		fofn 6756	. . . . . . 7 ⊢ (2nd :V–onto→V → 2nd Fn V)
23		elpreima 7012	. . . . . . 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 710	. . . . 5 ⊢ (𝑤 ∈ (◡2nd “ 𝐴) ↔ (2nd ‘𝑤) ∈ 𝐴)
26	25	anbi1i 625	. . . 4 ⊢ ((𝑤 ∈ (◡2nd “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((2nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
27	18, 19, 26	3bitri 297	. . 3 ⊢ (𝑤 ∈ (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((2nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
28		elxp7 7978	. . 3 ⊢ (𝑤 ∈ (𝐵 × 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐴)))
29	16, 27, 28	3bitr4g 314	. 2 ⊢ (𝐴 ⊆ 𝐶 → (𝑤 ∈ (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐵 × 𝐴)))
30	29	eqrdv 2735	1 ⊢ (𝐴 ⊆ 𝐶 → (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903   × cxp 5630  ^◡ccnv 5631   ↾ cres 5634   “ cima 5635   Fn wfn 6495  –onto→wfo 6498  ‘cfv 6500  1^st c1st 7941  2^nd c2nd 7942

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 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-1st 7943  df-2nd 7944

This theorem is referenced by:  sxbrsigalem2  34463