Theorem 2ndpreima 32807

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 7973	. . . . . 6 ⊢ (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐶)))
2	1	anbi1i 630	. . . . 5 ⊢ ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐶)) ∧ (2nd ‘𝑤) ∈ 𝐴))
3		ssel 3916	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐶 → ((2nd ‘𝑤) ∈ 𝐴 → (2nd ‘𝑤) ∈ 𝐶))
4	3	pm4.71rd 567	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐶 → ((2nd ‘𝑤) ∈ 𝐴 ↔ ((2nd ‘𝑤) ∈ 𝐶 ∧ (2nd ‘𝑤) ∈ 𝐴)))
5	4	anbi2d 636	. . . . . 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 469	. . . . . . . 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 225	. . . . . . 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 469	. . . . . . . 8 ⊢ (((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ (2nd ‘𝑤) ∈ 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐶)))
10	9	anbi1i 630	. . . . . . 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 306	. . . . 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 291	. . . 4 ⊢ (𝐴 ⊆ 𝐶 → ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ (2nd ‘𝑤) ∈ 𝐴)))
14		ancom 461	. . . 4 ⊢ ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd ‘𝑤) ∈ 𝐴) ↔ ((2nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
15		anass 469	. . . 4 ⊢ (((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ (2nd ‘𝑤) ∈ 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐴)))
16	13, 14, 15	3bitr3g 314	. . 3 ⊢ (𝐴 ⊆ 𝐶 → (((2nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐴))))
17		cnvresima 6188	. . . . 5 ⊢ (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = ((◡2nd “ 𝐴) ∩ (𝐵 × 𝐶))
18	17	eleq2i 2832	. . . 4 ⊢ (𝑤 ∈ (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((◡2nd “ 𝐴) ∩ (𝐵 × 𝐶)))
19		elin 3906	. . . 4 ⊢ (𝑤 ∈ ((◡2nd “ 𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (◡2nd “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
20		vex 3436	. . . . . 6 ⊢ 𝑤 ∈ V
21		fo2nd 7959	. . . . . . 7 ⊢ 2nd :V–onto→V
22		fofn 6748	. . . . . . 7 ⊢ (2nd :V–onto→V → 2nd Fn V)
23		elpreima 7006	. . . . . . 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 715	. . . . 5 ⊢ (𝑤 ∈ (◡2nd “ 𝐴) ↔ (2nd ‘𝑤) ∈ 𝐴)
26	25	anbi1i 630	. . . 4 ⊢ ((𝑤 ∈ (◡2nd “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((2nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
27	18, 19, 26	3bitri 298	. . 3 ⊢ (𝑤 ∈ (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((2nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
28		elxp7 7973	. . 3 ⊢ (𝑤 ∈ (𝐵 × 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐴)))
29	16, 27, 28	3bitr4g 315	. 2 ⊢ (𝐴 ⊆ 𝐶 → (𝑤 ∈ (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐵 × 𝐴)))
30	29	eqrdv 2738	1 ⊢ (𝐴 ⊆ 𝐶 → (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432   ∩ cin 3889   ⊆ wss 3890   × cxp 5623  ^◡ccnv 5624   ↾ cres 5627   “ cima 5628   Fn wfn 6487  –onto→wfo 6490  ‘cfv 6492  1^st c1st 7936  2^nd c2nd 7937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1st 7938  df-2nd 7939

This theorem is referenced by:  sxbrsigalem2  34477