Theorem opelco3 31678

Description: Alternate way of saying that an ordered pair is in a composition. (Contributed by Scott Fenton, 6-May-2018.)

Assertion

Ref	Expression
opelco3	|- ( <. A , B >. e. ( C o. D ) <-> B e. ( C " ( D " { A } ) ) )

Proof of Theorem opelco3

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 4654	. 2 |- ( A ( C o. D ) B <-> <. A , B >. e. ( C o. D ) )
2		relco 5633	. . . 4 |- Rel ( C o. D )
3		brrelex12 5155	. . . 4 |- ( ( Rel ( C o. D ) /\ A ( C o. D ) B ) -> ( A e. _V /\ B e. _V ) )
4	2, 3	mpan 706	. . 3 |- ( A ( C o. D ) B -> ( A e. _V /\ B e. _V ) )
5		snprc 4253	. . . . . 6 |- ( -. A e. _V <-> { A } = (/) )
6		noel 3919	. . . . . . 7 |- -. B e. (/)
7		imaeq2 5462	. . . . . . . . . 10 |- ( { A } = (/) -> ( D " { A } ) = ( D " (/) ) )
8	7	imaeq2d 5466	. . . . . . . . 9 |- ( { A } = (/) -> ( C " ( D " { A } ) ) = ( C " ( D " (/) ) ) )
9		ima0 5481	. . . . . . . . . . 11 |- ( D " (/) ) = (/)
10	9	imaeq2i 5464	. . . . . . . . . 10 |- ( C " ( D " (/) ) ) = ( C " (/) )
11		ima0 5481	. . . . . . . . . 10 |- ( C " (/) ) = (/)
12	10, 11	eqtri 2644	. . . . . . . . 9 |- ( C " ( D " (/) ) ) = (/)
13	8, 12	syl6eq 2672	. . . . . . . 8 |- ( { A } = (/) -> ( C " ( D " { A } ) ) = (/) )
14	13	eleq2d 2687	. . . . . . 7 |- ( { A } = (/) -> ( B e. ( C " ( D " { A } ) ) <-> B e. (/) ) )
15	6, 14	mtbiri 317	. . . . . 6 |- ( { A } = (/) -> -. B e. ( C " ( D " { A } ) ) )
16	5, 15	sylbi 207	. . . . 5 |- ( -. A e. _V -> -. B e. ( C " ( D " { A } ) ) )
17	16	con4i 113	. . . 4 |- ( B e. ( C " ( D " { A } ) ) -> A e. _V )
18		elex 3212	. . . 4 |- ( B e. ( C " ( D " { A } ) ) -> B e. _V )
19	17, 18	jca 554	. . 3 |- ( B e. ( C " ( D " { A } ) ) -> ( A e. _V /\ B e. _V ) )
20		df-rex 2918	. . . . 5 |- ( E. z e. ( D " { A } ) z C B <-> E. z ( z e. ( D " { A } ) /\ z C B ) )
21		vex 3203	. . . . . . . . . 10 |- z e. _V
22		elimasng 5491	. . . . . . . . . 10 |- ( ( A e. _V /\ z e. _V ) -> ( z e. ( D " { A } ) <-> <. A , z >. e. D ) )
23	21, 22	mpan2 707	. . . . . . . . 9 |- ( A e. _V -> ( z e. ( D " { A } ) <-> <. A , z >. e. D ) )
24		df-br 4654	. . . . . . . . 9 |- ( A D z <-> <. A , z >. e. D )
25	23, 24	syl6bbr 278	. . . . . . . 8 |- ( A e. _V -> ( z e. ( D " { A } ) <-> A D z ) )
26	25	adantr 481	. . . . . . 7 |- ( ( A e. _V /\ B e. _V ) -> ( z e. ( D " { A } ) <-> A D z ) )
27	26	anbi1d 741	. . . . . 6 |- ( ( A e. _V /\ B e. _V ) -> ( ( z e. ( D " { A } ) /\ z C B ) <-> ( A D z /\ z C B ) ) )
28	27	exbidv 1850	. . . . 5 |- ( ( A e. _V /\ B e. _V ) -> ( E. z ( z e. ( D " { A } ) /\ z C B ) <-> E. z ( A D z /\ z C B ) ) )
29	20, 28	syl5rbb 273	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( E. z ( A D z /\ z C B ) <-> E. z e. ( D " { A } ) z C B ) )
30		brcog 5288	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( A ( C o. D ) B <-> E. z ( A D z /\ z C B ) ) )
31		elimag 5470	. . . . 5 |- ( B e. _V -> ( B e. ( C " ( D " { A } ) ) <-> E. z e. ( D " { A } ) z C B ) )
32	31	adantl 482	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( B e. ( C " ( D " { A } ) ) <-> E. z e. ( D " { A } ) z C B ) )
33	29, 30, 32	3bitr4d 300	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( A ( C o. D ) B <-> B e. ( C " ( D " { A } ) ) ) )
34	4, 19, 33	pm5.21nii 368	. 2 |- ( A ( C o. D ) B <-> B e. ( C " ( D " { A } ) ) )
35	1, 34	bitr3i 266	1 |- ( <. A , B >. e. ( C o. D ) <-> B e. ( C " ( D " { A } ) ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   (/)c0 3915   {csn 4177   <.cop 4183   class class class wbr 4653   "cima 5117   o. ccom 5118   Rel wrel 5119

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by: (None)