Theorem caofrss 6006

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 5555	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
3		caofcom.3	. . . . 5 |- ( ph -> G : A --> S )
4	3	ffvelrnda 5555	. . . 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 2514	. . . . 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 3932	. . . . . 6 |- ( x = ( F ` w ) -> ( x R y <-> ( F ` w ) R y ) )
9		breq1 3932	. . . . . 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 3933	. . . . . 6 |- ( y = ( G ` w ) -> ( ( F ` w ) R y <-> ( F ` w ) R ( G ` w ) ) )
12		breq2 3933	. . . . . 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 2803	. . . 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 1215	. . 3 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) R ( G ` w ) -> ( F ` w ) T ( G ` w ) ) )
16	15	ralimdva 2499	. 2 |- ( ph -> ( A. w e. A ( F ` w ) R ( G ` w ) -> A. w e. A ( F ` w ) T ( G ` w ) ) )
17		ffn 5272	. . . 4 |- ( F : A --> S -> F Fn A )
18	1, 17	syl 14	. . 3 |- ( ph -> F Fn A )
19		ffn 5272	. . . 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 3285	. . 3 |- ( A i^i A ) = A
23		eqidd 2140	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
24		eqidd 2140	. . 3 |- ( ( ph /\ w e. A ) -> ( G ` w ) = ( G ` w ) )
25	18, 20, 21, 21, 22, 23, 24	ofrfval 5990	. 2 |- ( ph -> ( F oR R G <-> A. w e. A ( F ` w ) R ( G ` w ) ) )
26	18, 20, 21, 21, 22, 23, 24	ofrfval 5990	. 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 1331   e. wcel 1480   A.wral 2416   class class class wbr 3929   Fn wfn 5118   -->wf 5119   ` cfv 5123   oRcofr 5981

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ofr 5983

This theorem is referenced by: (None)