Theorem mptmpt2opabbrd 7248

Description: The operation value of a function value of a collection of ordered pairs of elements related in two ways. (Contributed by Alexander van Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.)

Hypotheses

Ref	Expression
mptmpt2opabbrd.g	|- ( ph -> G e. W )
mptmpt2opabbrd.x	|- ( ph -> X e. ( A ` G ) )
mptmpt2opabbrd.y	|- ( ph -> Y e. ( B ` G ) )
mptmpt2opabbrd.v	|- ( ph -> { <. f , h >. | ps } e. V )
mptmpt2opabbrd.r	|- ( ( ph /\ f ( D ` G ) h ) -> ps )
mptmpt2opabbrd.1	|- ( ( a = X /\ b = Y ) -> ( ta <-> th ) )
mptmpt2opabbrd.2	|- ( g = G -> ( ch <-> ta ) )
mptmpt2opabbrd.m	|- M = ( g e. _V |-> ( a e. ( A ` g ) , b e. ( B ` g ) |-> { <. f , h >. | ( ch /\ f ( D ` g ) h ) } ) )

Assertion

Ref	Expression
mptmpt2opabbrd	|- ( ph -> ( X ( M ` G ) Y ) = { <. f , h >. | ( th /\ f ( D ` G ) h ) } )

Distinct variable groups:   A, a, b, g   B, a, b, g   D, a, b, g   G, a, b, f, g, h   g, W   X, a, b, f, g, h   Y, a, b, f, g, h   ph, f, h   ta, g   th, a, b

Allowed substitution hints:   ph( g, a, b)   ps( f, g, h, a, b)   ch( f, g, h, a, b)   th( f, g, h)   ta( f, h, a, b)   A( f, h)   B( f, h)   D( f, h)   M( f, g, h, a, b)   V( f, g, h, a, b)   W( f, h, a, b)

Proof of Theorem mptmpt2opabbrd

Step	Hyp	Ref	Expression
1		mptmpt2opabbrd.g	. . . 4 |- ( ph -> G e. W )
2		mptmpt2opabbrd.m	. . . . . 6 |- M = ( g e. _V |-> ( a e. ( A ` g ) , b e. ( B ` g ) |-> { <. f , h >. | ( ch /\ f ( D ` g ) h ) } ) )
3	2	a1i 11	. . . . 5 |- ( ( G e. W /\ G e. W ) -> M = ( g e. _V |-> ( a e. ( A ` g ) , b e. ( B ` g ) |-> { <. f , h >. | ( ch /\ f ( D ` g ) h ) } ) ) )
4		fveq2 6191	. . . . . . 7 |- ( g = G -> ( A ` g ) = ( A ` G ) )
5		fveq2 6191	. . . . . . 7 |- ( g = G -> ( B ` g ) = ( B ` G ) )
6		mptmpt2opabbrd.2	. . . . . . . . 9 |- ( g = G -> ( ch <-> ta ) )
7		fveq2 6191	. . . . . . . . . 10 |- ( g = G -> ( D ` g ) = ( D ` G ) )
8	7	breqd 4664	. . . . . . . . 9 |- ( g = G -> ( f ( D ` g ) h <-> f ( D ` G ) h ) )
9	6, 8	anbi12d 747	. . . . . . . 8 |- ( g = G -> ( ( ch /\ f ( D ` g ) h ) <-> ( ta /\ f ( D ` G ) h ) ) )
10	9	opabbidv 4716	. . . . . . 7 |- ( g = G -> { <. f , h >. | ( ch /\ f ( D ` g ) h ) } = { <. f , h >. | ( ta /\ f ( D ` G ) h ) } )
11	4, 5, 10	mpt2eq123dv 6717	. . . . . 6 |- ( g = G -> ( a e. ( A ` g ) , b e. ( B ` g ) |-> { <. f , h >. | ( ch /\ f ( D ` g ) h ) } ) = ( a e. ( A ` G ) , b e. ( B ` G ) |-> { <. f , h >. | ( ta /\ f ( D ` G ) h ) } ) )
12	11	adantl 482	. . . . 5 |- ( ( ( G e. W /\ G e. W ) /\ g = G ) -> ( a e. ( A ` g ) , b e. ( B ` g ) |-> { <. f , h >. | ( ch /\ f ( D ` g ) h ) } ) = ( a e. ( A ` G ) , b e. ( B ` G ) |-> { <. f , h >. | ( ta /\ f ( D ` G ) h ) } ) )
13		elex 3212	. . . . . 6 |- ( G e. W -> G e. _V )
14	13	adantr 481	. . . . 5 |- ( ( G e. W /\ G e. W ) -> G e. _V )
15		fvex 6201	. . . . . . 7 |- ( A ` G ) e. _V
16		fvex 6201	. . . . . . 7 |- ( B ` G ) e. _V
17	15, 16	pm3.2i 471	. . . . . 6 |- ( ( A ` G ) e. _V /\ ( B ` G ) e. _V )
18		mpt2exga 7246	. . . . . 6 |- ( ( ( A ` G ) e. _V /\ ( B ` G ) e. _V ) -> ( a e. ( A ` G ) , b e. ( B ` G ) |-> { <. f , h >. | ( ta /\ f ( D ` G ) h ) } ) e. _V )
19	17, 18	mp1i 13	. . . . 5 |- ( ( G e. W /\ G e. W ) -> ( a e. ( A ` G ) , b e. ( B ` G ) |-> { <. f , h >. | ( ta /\ f ( D ` G ) h ) } ) e. _V )
20	3, 12, 14, 19	fvmptd 6288	. . . 4 |- ( ( G e. W /\ G e. W ) -> ( M ` G ) = ( a e. ( A ` G ) , b e. ( B ` G ) |-> { <. f , h >. | ( ta /\ f ( D ` G ) h ) } ) )
21	1, 1, 20	syl2anc 693	. . 3 |- ( ph -> ( M ` G ) = ( a e. ( A ` G ) , b e. ( B ` G ) |-> { <. f , h >. | ( ta /\ f ( D ` G ) h ) } ) )
22	21	oveqd 6667	. 2 |- ( ph -> ( X ( M ` G ) Y ) = ( X ( a e. ( A ` G ) , b e. ( B ` G ) |-> { <. f , h >. | ( ta /\ f ( D ` G ) h ) } ) Y ) )
23		mptmpt2opabbrd.x	. . 3 |- ( ph -> X e. ( A ` G ) )
24		mptmpt2opabbrd.y	. . 3 |- ( ph -> Y e. ( B ` G ) )
25		ancom 466	. . . . 5 |- ( ( th /\ f ( D ` G ) h ) <-> ( f ( D ` G ) h /\ th ) )
26	25	opabbii 4717	. . . 4 |- { <. f , h >. | ( th /\ f ( D ` G ) h ) } = { <. f , h >. | ( f ( D ` G ) h /\ th ) }
27		mptmpt2opabbrd.r	. . . . 5 |- ( ( ph /\ f ( D ` G ) h ) -> ps )
28		mptmpt2opabbrd.v	. . . . 5 |- ( ph -> { <. f , h >. | ps } e. V )
29	27, 28	opabresex2d 6696	. . . 4 |- ( ph -> { <. f , h >. | ( f ( D ` G ) h /\ th ) } e. _V )
30	26, 29	syl5eqel 2705	. . 3 |- ( ph -> { <. f , h >. | ( th /\ f ( D ` G ) h ) } e. _V )
31		mptmpt2opabbrd.1	. . . . . 6 |- ( ( a = X /\ b = Y ) -> ( ta <-> th ) )
32	31	anbi1d 741	. . . . 5 |- ( ( a = X /\ b = Y ) -> ( ( ta /\ f ( D ` G ) h ) <-> ( th /\ f ( D ` G ) h ) ) )
33	32	opabbidv 4716	. . . 4 |- ( ( a = X /\ b = Y ) -> { <. f , h >. | ( ta /\ f ( D ` G ) h ) } = { <. f , h >. | ( th /\ f ( D ` G ) h ) } )
34		eqid 2622	. . . 4 |- ( a e. ( A ` G ) , b e. ( B ` G ) |-> { <. f , h >. | ( ta /\ f ( D ` G ) h ) } ) = ( a e. ( A ` G ) , b e. ( B ` G ) |-> { <. f , h >. | ( ta /\ f ( D ` G ) h ) } )
35	33, 34	ovmpt2ga 6790	. . 3 |- ( ( X e. ( A ` G ) /\ Y e. ( B ` G ) /\ { <. f , h >. | ( th /\ f ( D ` G ) h ) } e. _V ) -> ( X ( a e. ( A ` G ) , b e. ( B ` G ) |-> { <. f , h >. | ( ta /\ f ( D ` G ) h ) } ) Y ) = { <. f , h >. | ( th /\ f ( D ` G ) h ) } )
36	23, 24, 30, 35	syl3anc 1326	. 2 |- ( ph -> ( X ( a e. ( A ` G ) , b e. ( B ` G ) |-> { <. f , h >. | ( ta /\ f ( D ` G ) h ) } ) Y ) = { <. f , h >. | ( th /\ f ( D ` G ) h ) } )
37	22, 36	eqtrd 2656	1 |- ( ph -> ( X ( M ` G ) Y ) = { <. f , h >. | ( th /\ f ( D ` G ) h ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   {copab 4712   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652

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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  mptmpt2opabovd  7249  wlkson  26552