Theorem caofrss 6930

Description: Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

Ref	Expression
caofref.1	|- ( ph -> A e. V )
caofref.2	|- ( ph -> F : A --> S )
caofcom.3	|- ( ph -> G : A --> S )
caofrss.4	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x R y -> x T y ) )

Assertion

Ref	Expression
caofrss	|- ( ph -> ( F oR R G -> F oR T G ) )

Distinct variable groups:   x, y, F   x, G, y   ph, x, y   x, R, y   x, S, y   x, T, y

Allowed substitution hints:   A( x, y)   V( x, y)

Proof of Theorem caofrss

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofref.2	. . . . 5 |- ( ph -> F : A --> S )
2	1	ffvelrnda 6359	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
3		caofcom.3	. . . . 5 |- ( ph -> G : A --> S )
4	3	ffvelrnda 6359	. . . 4 |- ( ( ph /\ w e. A ) -> ( G ` w ) e. S )
5		caofrss.4	. . . . . 6 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x R y -> x T y ) )
6	5	ralrimivva 2971	. . . . 5 |- ( ph -> A. x e. S A. y e. S ( x R y -> x T y ) )
7	6	adantr 481	. . . 4 |- ( ( ph /\ w e. A ) -> A. x e. S A. y e. S ( x R y -> x T y ) )
8		breq1 4656	. . . . . 6 |- ( x = ( F ` w ) -> ( x R y <-> ( F ` w ) R y ) )
9		breq1 4656	. . . . . 6 |- ( x = ( F ` w ) -> ( x T y <-> ( F ` w ) T y ) )
10	8, 9	imbi12d 334	. . . . 5 |- ( x = ( F ` w ) -> ( ( x R y -> x T y ) <-> ( ( F ` w ) R y -> ( F ` w ) T y ) ) )
11		breq2 4657	. . . . . 6 |- ( y = ( G ` w ) -> ( ( F ` w ) R y <-> ( F ` w ) R ( G ` w ) ) )
12		breq2 4657	. . . . . 6 |- ( y = ( G ` w ) -> ( ( F ` w ) T y <-> ( F ` w ) T ( G ` w ) ) )
13	11, 12	imbi12d 334	. . . . 5 |- ( y = ( G ` w ) -> ( ( ( F ` w ) R y -> ( F ` w ) T y ) <-> ( ( F ` w ) R ( G ` w ) -> ( F ` w ) T ( G ` w ) ) ) )
14	10, 13	rspc2va 3323	. . . 4 |- ( ( ( ( F ` w ) e. S /\ ( G ` w ) e. S ) /\ A. x e. S A. y e. S ( x R y -> x T y ) ) -> ( ( F ` w ) R ( G ` w ) -> ( F ` w ) T ( G ` w ) ) )
15	2, 4, 7, 14	syl21anc 1325	. . 3 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) R ( G ` w ) -> ( F ` w ) T ( G ` w ) ) )
16	15	ralimdva 2962	. 2 |- ( ph -> ( A. w e. A ( F ` w ) R ( G ` w ) -> A. w e. A ( F ` w ) T ( G ` w ) ) )
17		ffn 6045	. . . 4 |- ( F : A --> S -> F Fn A )
18	1, 17	syl 17	. . 3 |- ( ph -> F Fn A )
19		ffn 6045	. . . 4 |- ( G : A --> S -> G Fn A )
20	3, 19	syl 17	. . 3 |- ( ph -> G Fn A )
21		caofref.1	. . 3 |- ( ph -> A e. V )
22		inidm 3822	. . 3 |- ( A i^i A ) = A
23		eqidd 2623	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
24		eqidd 2623	. . 3 |- ( ( ph /\ w e. A ) -> ( G ` w ) = ( G ` w ) )
25	18, 20, 21, 21, 22, 23, 24	ofrfval 6905	. 2 |- ( ph -> ( F oR R G <-> A. w e. A ( F ` w ) R ( G ` w ) ) )
26	18, 20, 21, 21, 22, 23, 24	ofrfval 6905	. 2 |- ( ph -> ( F oR T G <-> A. w e. A ( F ` w ) T ( G ` w ) ) )
27	16, 25, 26	3imtr4d 283	1 |- ( ph -> ( F oR R G -> F oR T G ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   Fn wfn 5883   -->wf 5884   ` cfv 5888   oRcofr 6896

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-rep 4771  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-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-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-ofr 6898

This theorem is referenced by: (None)