Theorem fnopabco 33517

Description: Composition of a function with a function abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)

Hypotheses

Ref	Expression
fnopabco.1	|- ( x e. A -> B e. C )
fnopabco.2	|- F = { <. x , y >. | ( x e. A /\ y = B ) }
fnopabco.3	|- G = { <. x , y >. | ( x e. A /\ y = ( H ` B ) ) }

Assertion

Ref	Expression
fnopabco	|- ( H Fn C -> G = ( H o. F ) )

Distinct variable groups:   x, C, y   y, B   x, H, y   x, A, y

Allowed substitution hints:   B( x)   F( x, y)   G( x, y)

Proof of Theorem fnopabco

Step	Hyp	Ref	Expression
1		fnopabco.1	. . . 4 |- ( x e. A -> B e. C )
2	1	adantl 482	. . 3 |- ( ( H Fn C /\ x e. A ) -> B e. C )
3		fnopabco.2	. . . . 5 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
4		df-mpt 4730	. . . . 5 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
5	3, 4	eqtr4i 2647	. . . 4 |- F = ( x e. A |-> B )
6	5	a1i 11	. . 3 |- ( H Fn C -> F = ( x e. A |-> B ) )
7		dffn5 6241	. . . 4 |- ( H Fn C <-> H = ( y e. C |-> ( H ` y ) ) )
8	7	biimpi 206	. . 3 |- ( H Fn C -> H = ( y e. C |-> ( H ` y ) ) )
9		fveq2 6191	. . 3 |- ( y = B -> ( H ` y ) = ( H ` B ) )
10	2, 6, 8, 9	fmptco 6396	. 2 |- ( H Fn C -> ( H o. F ) = ( x e. A |-> ( H ` B ) ) )
11		fnopabco.3	. . 3 |- G = { <. x , y >. | ( x e. A /\ y = ( H ` B ) ) }
12		df-mpt 4730	. . 3 |- ( x e. A |-> ( H ` B ) ) = { <. x , y >. | ( x e. A /\ y = ( H ` B ) ) }
13	11, 12	eqtr4i 2647	. 2 |- G = ( x e. A |-> ( H ` B ) )
14	10, 13	syl6reqr 2675	1 |- ( H Fn C -> G = ( H o. F ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {copab 4712   |-> cmpt 4729   o. ccom 5118   Fn wfn 5883   ` 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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  opropabco  33518