Theorem 2ndpreima 31148

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 7911	. . . . . 6 ⊢ (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐶)))
2	1	anbi1i 624	. . . . 5 ⊢ ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd ‘𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐶)) ∧ (2nd ‘𝑤) ∈ 𝐴))
3		ssel 3924	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐶 → ((2nd ‘𝑤) ∈ 𝐴 → (2nd ‘𝑤) ∈ 𝐶))
4	3	pm4.71rd 563	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐶 → ((2nd ‘𝑤) ∈ 𝐴 ↔ ((2nd ‘𝑤) ∈ 𝐶 ∧ (2nd ‘𝑤) ∈ 𝐴)))
5	4	anbi2d 629	. . . . . 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 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 469	. . . . . . . 8 ⊢ (((𝑤 ∈ (V × V) ∧ (1st ‘𝑤) ∈ 𝐵) ∧ (2nd ‘𝑤) ∈ 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐶)))
10	9	anbi1i 624	. . . . . . 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 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 312	. . 3 ⊢ (𝐴 ⊆ 𝐶 → (((2nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) ∈ 𝐵 ∧ (2nd ‘𝑤) ∈ 𝐴))))
17		cnvresima 6155	. . . . 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 3913	. . . 4 ⊢ (𝑤 ∈ ((◡2nd “ 𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (◡2nd “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
20		vex 3445	. . . . . 6 ⊢ 𝑤 ∈ V
21		fo2nd 7897	. . . . . . 7 ⊢ 2nd :V–onto→V
22		fofn 6727	. . . . . . 7 ⊢ (2nd :V–onto→V → 2nd Fn V)
23		elpreima 6974	. . . . . . 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 706	. . . . 5 ⊢ (𝑤 ∈ (◡2nd “ 𝐴) ↔ (2nd ‘𝑤) ∈ 𝐴)
26	25	anbi1i 624	. . . 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 7911	. . 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 2735	1 ⊢ (𝐴 ⊆ 𝐶 → (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  Vcvv 3441   ∩ cin 3896   ⊆ wss 3897   × cxp 5605  ^◡ccnv 5606   ↾ cres 5609   “ cima 5610   Fn wfn 6460  –onto→wfo 6463  ‘cfv 6465  1^st c1st 7874  2^nd c2nd 7875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pr 5367  ax-un 7628

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-opab 5150  df-mpt 5171  df-id 5507  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-fo 6471  df-fv 6473  df-1st 7876  df-2nd 7877

This theorem is referenced by:  sxbrsigalem2  32359