Theorem 2ndpreima 32717

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 8049	. . . . . 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 3977	. . . . . . . 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 630	. . . . . 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 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 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 6250	. . . . 5 ⊢ (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = ((◡2nd “ 𝐴) ∩ (𝐵 × 𝐶))
18	17	eleq2i 2833	. . . 4 ⊢ (𝑤 ∈ (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((◡2nd “ 𝐴) ∩ (𝐵 × 𝐶)))
19		elin 3967	. . . 4 ⊢ (𝑤 ∈ ((◡2nd “ 𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (◡2nd “ 𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
20		vex 3484	. . . . . 6 ⊢ 𝑤 ∈ V
21		fo2nd 8035	. . . . . . 7 ⊢ 2nd :V–onto→V
22		fofn 6822	. . . . . . 7 ⊢ (2nd :V–onto→V → 2nd Fn V)
23		elpreima 7078	. . . . . . 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 709	. . . . 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 297	. . 3 ⊢ (𝑤 ∈ (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((2nd ‘𝑤) ∈ 𝐴 ∧ 𝑤 ∈ (𝐵 × 𝐶)))
28		elxp7 8049	. . 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 1540   ∈ wcel 2108  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951   × cxp 5683  ^◡ccnv 5684   ↾ cres 5687   “ cima 5688   Fn wfn 6556  –onto→wfo 6559  ‘cfv 6561  1^st c1st 8012  2^nd c2nd 8013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015

This theorem is referenced by:  sxbrsigalem2  34288