Theorem caofrss 6089

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	ffvelcdmda 5635	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
3		caofcom.3	. . . . 5 |- ( ph -> G : A --> S )
4	3	ffvelcdmda 5635	. . . 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 2553	. . . . 5 |- ( ph -> A. x e. S A. y e. S ( x R y -> x T y ) )
7	6	adantr 274	. . . 4 |- ( ( ph /\ w e. A ) -> A. x e. S A. y e. S ( x R y -> x T y ) )
8		breq1 3993	. . . . . 6 |- ( x = ( F ` w ) -> ( x R y <-> ( F ` w ) R y ) )
9		breq1 3993	. . . . . 6 |- ( x = ( F ` w ) -> ( x T y <-> ( F ` w ) T y ) )
10	8, 9	imbi12d 233	. . . . 5 |- ( x = ( F ` w ) -> ( ( x R y -> x T y ) <-> ( ( F ` w ) R y -> ( F ` w ) T y ) ) )
11		breq2 3994	. . . . . 6 |- ( y = ( G ` w ) -> ( ( F ` w ) R y <-> ( F ` w ) R ( G ` w ) ) )
12		breq2 3994	. . . . . 6 |- ( y = ( G ` w ) -> ( ( F ` w ) T y <-> ( F ` w ) T ( G ` w ) ) )
13	11, 12	imbi12d 233	. . . . 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 2849	. . . 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 1233	. . 3 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) R ( G ` w ) -> ( F ` w ) T ( G ` w ) ) )
16	15	ralimdva 2538	. 2 |- ( ph -> ( A. w e. A ( F ` w ) R ( G ` w ) -> A. w e. A ( F ` w ) T ( G ` w ) ) )
17		ffn 5349	. . . 4 |- ( F : A --> S -> F Fn A )
18	1, 17	syl 14	. . 3 |- ( ph -> F Fn A )
19		ffn 5349	. . . 4 |- ( G : A --> S -> G Fn A )
20	3, 19	syl 14	. . 3 |- ( ph -> G Fn A )
21		caofref.1	. . 3 |- ( ph -> A e. V )
22		inidm 3337	. . 3 |- ( A i^i A ) = A
23		eqidd 2172	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
24		eqidd 2172	. . 3 |- ( ( ph /\ w e. A ) -> ( G ` w ) = ( G ` w ) )
25	18, 20, 21, 21, 22, 23, 24	ofrfval 6073	. 2 |- ( ph -> ( F oR R G <-> A. w e. A ( F ` w ) R ( G ` w ) ) )
26	18, 20, 21, 21, 22, 23, 24	ofrfval 6073	. 2 |- ( ph -> ( F oR T G <-> A. w e. A ( F ` w ) T ( G ` w ) ) )
27	16, 25, 26	3imtr4d 202	1 |- ( ph -> ( F oR R G -> F oR T G ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1349   e. wcel 2142   A.wral 2449   class class class wbr 3990   Fn wfn 5195   -->wf 5196   ` cfv 5200   oRcofr 6064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-pow 4161  ax-pr 4195

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-reu 2456  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-id 4279  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-ofr 6066

This theorem is referenced by: (None)