Theorem fvproj 29899

Description: Value of a function on pairs, given two projections F and G. (Contributed by Thierry Arnoux, 30-Dec-2019.)

Hypotheses

Ref	Expression
fvproj.h	|- H = ( x e. A , y e. B |-> <. ( F ` x ) , ( G ` y ) >. )
fvproj.x	|- ( ph -> X e. A )
fvproj.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
fvproj	|- ( ph -> ( H ` <. X , Y >. ) = <. ( F ` X ) , ( G ` Y ) >. )

Distinct variable groups:   x, A, y   x, B, y   x, F, y   x, G, y

Allowed substitution hints:   ph( x, y)   H( x, y)   X( x, y)   Y( x, y)

Proof of Theorem fvproj

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 6653	. 2 |- ( X H Y ) = ( H ` <. X , Y >. )
2		fvproj.x	. . 3 |- ( ph -> X e. A )
3		fvproj.y	. . 3 |- ( ph -> Y e. B )
4		fveq2 6191	. . . . 5 |- ( a = X -> ( F ` a ) = ( F ` X ) )
5	4	opeq1d 4408	. . . 4 |- ( a = X -> <. ( F ` a ) , ( G ` b ) >. = <. ( F ` X ) , ( G ` b ) >. )
6		fveq2 6191	. . . . 5 |- ( b = Y -> ( G ` b ) = ( G ` Y ) )
7	6	opeq2d 4409	. . . 4 |- ( b = Y -> <. ( F ` X ) , ( G ` b ) >. = <. ( F ` X ) , ( G ` Y ) >. )
8		fvproj.h	. . . . 5 |- H = ( x e. A , y e. B |-> <. ( F ` x ) , ( G ` y ) >. )
9		fveq2 6191	. . . . . . 7 |- ( x = a -> ( F ` x ) = ( F ` a ) )
10	9	opeq1d 4408	. . . . . 6 |- ( x = a -> <. ( F ` x ) , ( G ` y ) >. = <. ( F ` a ) , ( G ` y ) >. )
11		fveq2 6191	. . . . . . 7 |- ( y = b -> ( G ` y ) = ( G ` b ) )
12	11	opeq2d 4409	. . . . . 6 |- ( y = b -> <. ( F ` a ) , ( G ` y ) >. = <. ( F ` a ) , ( G ` b ) >. )
13	10, 12	cbvmpt2v 6735	. . . . 5 |- ( x e. A , y e. B |-> <. ( F ` x ) , ( G ` y ) >. ) = ( a e. A , b e. B |-> <. ( F ` a ) , ( G ` b ) >. )
14	8, 13	eqtri 2644	. . . 4 |- H = ( a e. A , b e. B |-> <. ( F ` a ) , ( G ` b ) >. )
15		opex 4932	. . . 4 |- <. ( F ` X ) , ( G ` Y ) >. e. _V
16	5, 7, 14, 15	ovmpt2 6796	. . 3 |- ( ( X e. A /\ Y e. B ) -> ( X H Y ) = <. ( F ` X ) , ( G ` Y ) >. )
17	2, 3, 16	syl2anc 693	. 2 |- ( ph -> ( X H Y ) = <. ( F ` X ) , ( G ` Y ) >. )
18	1, 17	syl5eqr 2670	1 |- ( ph -> ( H ` <. X , Y >. ) = <. ( F ` X ) , ( G ` Y ) >. )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   <.cop 4183   ` 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-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-uni 4437  df-br 4654  df-opab 4713  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  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  fimaproj  29900  qtophaus  29903