Theorem caofrss 6203

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 5728	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
3		caofcom.3	. . . . 5 |- ( ph -> G : A --> S )
4	3	ffvelcdmda 5728	. . . 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 2589	. . . . 5 |- ( ph -> A. x e. S A. y e. S ( x R y -> x T y ) )
7	6	adantr 276	. . . 4 |- ( ( ph /\ w e. A ) -> A. x e. S A. y e. S ( x R y -> x T y ) )
8		breq1 4054	. . . . . 6 |- ( x = ( F ` w ) -> ( x R y <-> ( F ` w ) R y ) )
9		breq1 4054	. . . . . 6 |- ( x = ( F ` w ) -> ( x T y <-> ( F ` w ) T y ) )
10	8, 9	imbi12d 234	. . . . 5 |- ( x = ( F ` w ) -> ( ( x R y -> x T y ) <-> ( ( F ` w ) R y -> ( F ` w ) T y ) ) )
11		breq2 4055	. . . . . 6 |- ( y = ( G ` w ) -> ( ( F ` w ) R y <-> ( F ` w ) R ( G ` w ) ) )
12		breq2 4055	. . . . . 6 |- ( y = ( G ` w ) -> ( ( F ` w ) T y <-> ( F ` w ) T ( G ` w ) ) )
13	11, 12	imbi12d 234	. . . . 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 2895	. . . 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 1249	. . 3 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) R ( G ` w ) -> ( F ` w ) T ( G ` w ) ) )
16	15	ralimdva 2574	. 2 |- ( ph -> ( A. w e. A ( F ` w ) R ( G ` w ) -> A. w e. A ( F ` w ) T ( G ` w ) ) )
17		ffn 5435	. . . 4 |- ( F : A --> S -> F Fn A )
18	1, 17	syl 14	. . 3 |- ( ph -> F Fn A )
19		ffn 5435	. . . 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 3386	. . 3 |- ( A i^i A ) = A
23		eqidd 2207	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
24		eqidd 2207	. . 3 |- ( ( ph /\ w e. A ) -> ( G ` w ) = ( G ` w ) )
25	18, 20, 21, 21, 22, 23, 24	ofrfval 6180	. 2 |- ( ph -> ( F oR R G <-> A. w e. A ( F ` w ) R ( G ` w ) ) )
26	18, 20, 21, 21, 22, 23, 24	ofrfval 6180	. 2 |- ( ph -> ( F oR T G <-> A. w e. A ( F ` w ) T ( G ` w ) ) )
27	16, 25, 26	3imtr4d 203	1 |- ( ph -> ( F oR R G -> F oR T G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   A.wral 2485   class class class wbr 4051   Fn wfn 5275   -->wf 5276   ` cfv 5280   oRcofr 6170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ofr 6172

This theorem is referenced by: (None)