Theorem fvmptopab 6697

Description: The function value of a mapping M to a restricted binary relation expressed as an ordered-pair class abstraction: The restricted binary relation is a binary relation given as value of a function F restricted by the condition ps. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.)

Hypotheses

Ref	Expression
fvmptopab.1	|- ( ( ph /\ z = Z ) -> ( ch <-> ps ) )
fvmptopab.2	|- ( ph -> { <. x , y >. | x ( F ` Z ) y } e. _V )
fvmptopab.3	|- M = ( z e. _V |-> { <. x , y >. | ( x ( F ` z ) y /\ ch ) } )

Assertion

Ref	Expression
fvmptopab	|- ( ph -> ( M ` Z ) = { <. x , y >. | ( x ( F ` Z ) y /\ ps ) } )

Distinct variable groups:   z, F   x, Z, y, z   ph, x, y, z   ps, z

Allowed substitution hints:   ps( x, y)   ch( x, y, z)   F( x, y)   M( x, y, z)

Proof of Theorem fvmptopab

Step	Hyp	Ref	Expression
1		fvmptopab.3	. . . . 5 |- M = ( z e. _V |-> { <. x , y >. | ( x ( F ` z ) y /\ ch ) } )
2	1	a1i 11	. . . 4 |- ( ( Z e. _V /\ ph ) -> M = ( z e. _V |-> { <. x , y >. | ( x ( F ` z ) y /\ ch ) } ) )
3		fveq2 6191	. . . . . . . 8 |- ( z = Z -> ( F ` z ) = ( F ` Z ) )
4	3	breqd 4664	. . . . . . 7 |- ( z = Z -> ( x ( F ` z ) y <-> x ( F ` Z ) y ) )
5	4	adantl 482	. . . . . 6 |- ( ( ( Z e. _V /\ ph ) /\ z = Z ) -> ( x ( F ` z ) y <-> x ( F ` Z ) y ) )
6		fvmptopab.1	. . . . . . 7 |- ( ( ph /\ z = Z ) -> ( ch <-> ps ) )
7	6	adantll 750	. . . . . 6 |- ( ( ( Z e. _V /\ ph ) /\ z = Z ) -> ( ch <-> ps ) )
8	5, 7	anbi12d 747	. . . . 5 |- ( ( ( Z e. _V /\ ph ) /\ z = Z ) -> ( ( x ( F ` z ) y /\ ch ) <-> ( x ( F ` Z ) y /\ ps ) ) )
9	8	opabbidv 4716	. . . 4 |- ( ( ( Z e. _V /\ ph ) /\ z = Z ) -> { <. x , y >. | ( x ( F ` z ) y /\ ch ) } = { <. x , y >. | ( x ( F ` Z ) y /\ ps ) } )
10		simpl 473	. . . 4 |- ( ( Z e. _V /\ ph ) -> Z e. _V )
11		id 22	. . . . . 6 |- ( x ( F ` Z ) y -> x ( F ` Z ) y )
12	11	gen2 1723	. . . . 5 |- A. x A. y ( x ( F ` Z ) y -> x ( F ` Z ) y )
13		fvmptopab.2	. . . . . 6 |- ( ph -> { <. x , y >. | x ( F ` Z ) y } e. _V )
14	13	adantl 482	. . . . 5 |- ( ( Z e. _V /\ ph ) -> { <. x , y >. | x ( F ` Z ) y } e. _V )
15		opabbrex 6695	. . . . 5 |- ( ( A. x A. y ( x ( F ` Z ) y -> x ( F ` Z ) y ) /\ { <. x , y >. | x ( F ` Z ) y } e. _V ) -> { <. x , y >. | ( x ( F ` Z ) y /\ ps ) } e. _V )
16	12, 14, 15	sylancr 695	. . . 4 |- ( ( Z e. _V /\ ph ) -> { <. x , y >. | ( x ( F ` Z ) y /\ ps ) } e. _V )
17	2, 9, 10, 16	fvmptd 6288	. . 3 |- ( ( Z e. _V /\ ph ) -> ( M ` Z ) = { <. x , y >. | ( x ( F ` Z ) y /\ ps ) } )
18	17	ex 450	. 2 |- ( Z e. _V -> ( ph -> ( M ` Z ) = { <. x , y >. | ( x ( F ` Z ) y /\ ps ) } ) )
19		fvprc 6185	. . . 4 |- ( -. Z e. _V -> ( M ` Z ) = (/) )
20		br0 4701	. . . . . . . 8 |- -. x (/) y
21		fvprc 6185	. . . . . . . . 9 |- ( -. Z e. _V -> ( F ` Z ) = (/) )
22	21	breqd 4664	. . . . . . . 8 |- ( -. Z e. _V -> ( x ( F ` Z ) y <-> x (/) y ) )
23	20, 22	mtbiri 317	. . . . . . 7 |- ( -. Z e. _V -> -. x ( F ` Z ) y )
24	23	intnanrd 963	. . . . . 6 |- ( -. Z e. _V -> -. ( x ( F ` Z ) y /\ ps ) )
25	24	alrimivv 1856	. . . . 5 |- ( -. Z e. _V -> A. x A. y -. ( x ( F ` Z ) y /\ ps ) )
26		opab0 5007	. . . . 5 |- ( { <. x , y >. | ( x ( F ` Z ) y /\ ps ) } = (/) <-> A. x A. y -. ( x ( F ` Z ) y /\ ps ) )
27	25, 26	sylibr 224	. . . 4 |- ( -. Z e. _V -> { <. x , y >. | ( x ( F ` Z ) y /\ ps ) } = (/) )
28	19, 27	eqtr4d 2659	. . 3 |- ( -. Z e. _V -> ( M ` Z ) = { <. x , y >. | ( x ( F ` Z ) y /\ ps ) } )
29	28	a1d 25	. 2 |- ( -. Z e. _V -> ( ph -> ( M ` Z ) = { <. x , y >. | ( x ( F ` Z ) y /\ ps ) } ) )
30	18, 29	pm2.61i 176	1 |- ( ph -> ( M ` Z ) = { <. x , y >. | ( x ( F ` Z ) y /\ ps ) } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   {copab 4712   |-> cmpt 4729   ` cfv 5888

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

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-csb 3534  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-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  trlsfval  26592  pthsfval  26617  spthsfval  26618  clwlks  26668  crcts  26683  cycls  26684  eupths  27060