Theorem 2ndpreima 29485

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

Assertion

Ref	Expression
2ndpreima	|- ( A C_ C -> ( `' ( 2nd |` ( B X. C ) ) " A ) = ( B X. A ) )

Proof of Theorem 2ndpreima

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3597	. . . . . . . 8 |- ( A C_ C -> ( ( 2nd ` w ) e. A -> ( 2nd ` w ) e. C ) )
2	1	pm4.71rd 667	. . . . . . 7 |- ( A C_ C -> ( ( 2nd ` w ) e. A <-> ( ( 2nd ` w ) e. C /\ ( 2nd ` w ) e. A ) ) )
3	2	anbi2d 740	. . . . . 6 |- ( A C_ C -> ( ( ( w e. ( _V X. _V ) /\ ( 1st ` w ) e. B ) /\ ( 2nd ` w ) e. A ) <-> ( ( w e. ( _V X. _V ) /\ ( 1st ` w ) e. B ) /\ ( ( 2nd ` w ) e. C /\ ( 2nd ` w ) e. A ) ) ) )
4		anass 681	. . . . . . . 8 |- ( ( ( ( w e. ( _V X. _V ) /\ ( 1st ` w ) e. B ) /\ ( 2nd ` w ) e. C ) /\ ( 2nd ` w ) e. A ) <-> ( ( w e. ( _V X. _V ) /\ ( 1st ` w ) e. B ) /\ ( ( 2nd ` w ) e. C /\ ( 2nd ` w ) e. A ) ) )
5	4	bicomi 214	. . . . . . 7 |- ( ( ( w e. ( _V X. _V ) /\ ( 1st ` w ) e. B ) /\ ( ( 2nd ` w ) e. C /\ ( 2nd ` w ) e. A ) ) <-> ( ( ( w e. ( _V X. _V ) /\ ( 1st ` w ) e. B ) /\ ( 2nd ` w ) e. C ) /\ ( 2nd ` w ) e. A ) )
6	5	a1i 11	. . . . . 6 |- ( A C_ C -> ( ( ( w e. ( _V X. _V ) /\ ( 1st ` w ) e. B ) /\ ( ( 2nd ` w ) e. C /\ ( 2nd ` w ) e. A ) ) <-> ( ( ( w e. ( _V X. _V ) /\ ( 1st ` w ) e. B ) /\ ( 2nd ` w ) e. C ) /\ ( 2nd ` w ) e. A ) ) )
7		anass 681	. . . . . . . 8 |- ( ( ( w e. ( _V X. _V ) /\ ( 1st ` w ) e. B ) /\ ( 2nd ` w ) e. C ) <-> ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. C ) ) )
8	7	anbi1i 731	. . . . . . 7 |- ( ( ( ( w e. ( _V X. _V ) /\ ( 1st ` w ) e. B ) /\ ( 2nd ` w ) e. C ) /\ ( 2nd ` w ) e. A ) <-> ( ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. C ) ) /\ ( 2nd ` w ) e. A ) )
9	8	a1i 11	. . . . . 6 |- ( A C_ C -> ( ( ( ( w e. ( _V X. _V ) /\ ( 1st ` w ) e. B ) /\ ( 2nd ` w ) e. C ) /\ ( 2nd ` w ) e. A ) <-> ( ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. C ) ) /\ ( 2nd ` w ) e. A ) ) )
10	3, 6, 9	3bitrd 294	. . . . 5 |- ( A C_ C -> ( ( ( w e. ( _V X. _V ) /\ ( 1st ` w ) e. B ) /\ ( 2nd ` w ) e. A ) <-> ( ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. C ) ) /\ ( 2nd ` w ) e. A ) ) )
11		elxp7 7201	. . . . . 6 |- ( w e. ( B X. C ) <-> ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. C ) ) )
12	11	anbi1i 731	. . . . 5 |- ( ( w e. ( B X. C ) /\ ( 2nd ` w ) e. A ) <-> ( ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. C ) ) /\ ( 2nd ` w ) e. A ) )
13	10, 12	syl6rbbr 279	. . . 4 |- ( A C_ C -> ( ( w e. ( B X. C ) /\ ( 2nd ` w ) e. A ) <-> ( ( w e. ( _V X. _V ) /\ ( 1st ` w ) e. B ) /\ ( 2nd ` w ) e. A ) ) )
14		ancom 466	. . . 4 |- ( ( w e. ( B X. C ) /\ ( 2nd ` w ) e. A ) <-> ( ( 2nd ` w ) e. A /\ w e. ( B X. C ) ) )
15		anass 681	. . . 4 |- ( ( ( w e. ( _V X. _V ) /\ ( 1st ` w ) e. B ) /\ ( 2nd ` w ) e. A ) <-> ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. A ) ) )
16	13, 14, 15	3bitr3g 302	. . 3 |- ( A C_ C -> ( ( ( 2nd ` w ) e. A /\ w e. ( B X. C ) ) <-> ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. A ) ) ) )
17		cnvresima 5623	. . . . 5 |- ( `' ( 2nd |` ( B X. C ) ) " A ) = ( ( `' 2nd " A ) i^i ( B X. C ) )
18	17	eleq2i 2693	. . . 4 |- ( w e. ( `' ( 2nd |` ( B X. C ) ) " A ) <-> w e. ( ( `' 2nd " A ) i^i ( B X. C ) ) )
19		elin 3796	. . . 4 |- ( w e. ( ( `' 2nd " A ) i^i ( B X. C ) ) <-> ( w e. ( `' 2nd " A ) /\ w e. ( B X. C ) ) )
20		vex 3203	. . . . . 6 |- w e. _V
21		fo2nd 7189	. . . . . . 7 |- 2nd : _V -onto-> _V
22		fofn 6117	. . . . . . 7 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
23		elpreima 6337	. . . . . . 7 |- ( 2nd Fn _V -> ( w e. ( `' 2nd " A ) <-> ( w e. _V /\ ( 2nd ` w ) e. A ) ) )
24	21, 22, 23	mp2b 10	. . . . . 6 |- ( w e. ( `' 2nd " A ) <-> ( w e. _V /\ ( 2nd ` w ) e. A ) )
25	20, 24	mpbiran 953	. . . . 5 |- ( w e. ( `' 2nd " A ) <-> ( 2nd ` w ) e. A )
26	25	anbi1i 731	. . . 4 |- ( ( w e. ( `' 2nd " A ) /\ w e. ( B X. C ) ) <-> ( ( 2nd ` w ) e. A /\ w e. ( B X. C ) ) )
27	18, 19, 26	3bitri 286	. . 3 |- ( w e. ( `' ( 2nd |` ( B X. C ) ) " A ) <-> ( ( 2nd ` w ) e. A /\ w e. ( B X. C ) ) )
28		elxp7 7201	. . 3 |- ( w e. ( B X. A ) <-> ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. A ) ) )
29	16, 27, 28	3bitr4g 303	. 2 |- ( A C_ C -> ( w e. ( `' ( 2nd |` ( B X. C ) ) " A ) <-> w e. ( B X. A ) ) )
30	29	eqrdv 2620	1 |- ( A C_ C -> ( `' ( 2nd |` ( B X. C ) ) " A ) = ( B X. A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   X. cxp 5112   `'ccnv 5113   |` cres 5116   "cima 5117   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  sxbrsigalem2  30348