Theorem caoftrn 6075

Description: Transfer a transitivity 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 )
caofass.4	|- ( ph -> H : A --> S )
caoftrn.5	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x R y /\ y T z ) -> x U z ) )

Assertion

Ref	Expression
caoftrn	|- ( ph -> ( ( F oR R G /\ G oR T H ) -> F oR U H ) )

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

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

Proof of Theorem caoftrn

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caoftrn.5	. . . . . 6 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x R y /\ y T z ) -> x U z ) )
2	1	ralrimivvva 2549	. . . . 5 |- ( ph -> A. x e. S A. y e. S A. z e. S ( ( x R y /\ y T z ) -> x U z ) )
3	2	adantr 274	. . . 4 |- ( ( ph /\ w e. A ) -> A. x e. S A. y e. S A. z e. S ( ( x R y /\ y T z ) -> x U z ) )
4		caofref.2	. . . . . 6 |- ( ph -> F : A --> S )
5	4	ffvelrnda 5620	. . . . 5 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
6		caofcom.3	. . . . . 6 |- ( ph -> G : A --> S )
7	6	ffvelrnda 5620	. . . . 5 |- ( ( ph /\ w e. A ) -> ( G ` w ) e. S )
8		caofass.4	. . . . . 6 |- ( ph -> H : A --> S )
9	8	ffvelrnda 5620	. . . . 5 |- ( ( ph /\ w e. A ) -> ( H ` w ) e. S )
10		breq1 3985	. . . . . . . 8 |- ( x = ( F ` w ) -> ( x R y <-> ( F ` w ) R y ) )
11	10	anbi1d 461	. . . . . . 7 |- ( x = ( F ` w ) -> ( ( x R y /\ y T z ) <-> ( ( F ` w ) R y /\ y T z ) ) )
12		breq1 3985	. . . . . . 7 |- ( x = ( F ` w ) -> ( x U z <-> ( F ` w ) U z ) )
13	11, 12	imbi12d 233	. . . . . 6 |- ( x = ( F ` w ) -> ( ( ( x R y /\ y T z ) -> x U z ) <-> ( ( ( F ` w ) R y /\ y T z ) -> ( F ` w ) U z ) ) )
14		breq2 3986	. . . . . . . 8 |- ( y = ( G ` w ) -> ( ( F ` w ) R y <-> ( F ` w ) R ( G ` w ) ) )
15		breq1 3985	. . . . . . . 8 |- ( y = ( G ` w ) -> ( y T z <-> ( G ` w ) T z ) )
16	14, 15	anbi12d 465	. . . . . . 7 |- ( y = ( G ` w ) -> ( ( ( F ` w ) R y /\ y T z ) <-> ( ( F ` w ) R ( G ` w ) /\ ( G ` w ) T z ) ) )
17	16	imbi1d 230	. . . . . 6 |- ( y = ( G ` w ) -> ( ( ( ( F ` w ) R y /\ y T z ) -> ( F ` w ) U z ) <-> ( ( ( F ` w ) R ( G ` w ) /\ ( G ` w ) T z ) -> ( F ` w ) U z ) ) )
18		breq2 3986	. . . . . . . 8 |- ( z = ( H ` w ) -> ( ( G ` w ) T z <-> ( G ` w ) T ( H ` w ) ) )
19	18	anbi2d 460	. . . . . . 7 |- ( z = ( H ` w ) -> ( ( ( F ` w ) R ( G ` w ) /\ ( G ` w ) T z ) <-> ( ( F ` w ) R ( G ` w ) /\ ( G ` w ) T ( H ` w ) ) ) )
20		breq2 3986	. . . . . . 7 |- ( z = ( H ` w ) -> ( ( F ` w ) U z <-> ( F ` w ) U ( H ` w ) ) )
21	19, 20	imbi12d 233	. . . . . 6 |- ( z = ( H ` w ) -> ( ( ( ( F ` w ) R ( G ` w ) /\ ( G ` w ) T z ) -> ( F ` w ) U z ) <-> ( ( ( F ` w ) R ( G ` w ) /\ ( G ` w ) T ( H ` w ) ) -> ( F ` w ) U ( H ` w ) ) ) )
22	13, 17, 21	rspc3v 2846	. . . . 5 |- ( ( ( F ` w ) e. S /\ ( G ` w ) e. S /\ ( H ` w ) e. S ) -> ( A. x e. S A. y e. S A. z e. S ( ( x R y /\ y T z ) -> x U z ) -> ( ( ( F ` w ) R ( G ` w ) /\ ( G ` w ) T ( H ` w ) ) -> ( F ` w ) U ( H ` w ) ) ) )
23	5, 7, 9, 22	syl3anc 1228	. . . 4 |- ( ( ph /\ w e. A ) -> ( A. x e. S A. y e. S A. z e. S ( ( x R y /\ y T z ) -> x U z ) -> ( ( ( F ` w ) R ( G ` w ) /\ ( G ` w ) T ( H ` w ) ) -> ( F ` w ) U ( H ` w ) ) ) )
24	3, 23	mpd 13	. . 3 |- ( ( ph /\ w e. A ) -> ( ( ( F ` w ) R ( G ` w ) /\ ( G ` w ) T ( H ` w ) ) -> ( F ` w ) U ( H ` w ) ) )
25	24	ralimdva 2533	. 2 |- ( ph -> ( A. w e. A ( ( F ` w ) R ( G ` w ) /\ ( G ` w ) T ( H ` w ) ) -> A. w e. A ( F ` w ) U ( H ` w ) ) )
26		ffn 5337	. . . . . 6 |- ( F : A --> S -> F Fn A )
27	4, 26	syl 14	. . . . 5 |- ( ph -> F Fn A )
28		ffn 5337	. . . . . 6 |- ( G : A --> S -> G Fn A )
29	6, 28	syl 14	. . . . 5 |- ( ph -> G Fn A )
30		caofref.1	. . . . 5 |- ( ph -> A e. V )
31		inidm 3331	. . . . 5 |- ( A i^i A ) = A
32		eqidd 2166	. . . . 5 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
33		eqidd 2166	. . . . 5 |- ( ( ph /\ w e. A ) -> ( G ` w ) = ( G ` w ) )
34	27, 29, 30, 30, 31, 32, 33	ofrfval 6058	. . . 4 |- ( ph -> ( F oR R G <-> A. w e. A ( F ` w ) R ( G ` w ) ) )
35		ffn 5337	. . . . . 6 |- ( H : A --> S -> H Fn A )
36	8, 35	syl 14	. . . . 5 |- ( ph -> H Fn A )
37		eqidd 2166	. . . . 5 |- ( ( ph /\ w e. A ) -> ( H ` w ) = ( H ` w ) )
38	29, 36, 30, 30, 31, 33, 37	ofrfval 6058	. . . 4 |- ( ph -> ( G oR T H <-> A. w e. A ( G ` w ) T ( H ` w ) ) )
39	34, 38	anbi12d 465	. . 3 |- ( ph -> ( ( F oR R G /\ G oR T H ) <-> ( A. w e. A ( F ` w ) R ( G ` w ) /\ A. w e. A ( G ` w ) T ( H ` w ) ) ) )
40		r19.26 2592	. . 3 |- ( A. w e. A ( ( F ` w ) R ( G ` w ) /\ ( G ` w ) T ( H ` w ) ) <-> ( A. w e. A ( F ` w ) R ( G ` w ) /\ A. w e. A ( G ` w ) T ( H ` w ) ) )
41	39, 40	bitr4di 197	. 2 |- ( ph -> ( ( F oR R G /\ G oR T H ) <-> A. w e. A ( ( F ` w ) R ( G ` w ) /\ ( G ` w ) T ( H ` w ) ) ) )
42	27, 36, 30, 30, 31, 32, 37	ofrfval 6058	. 2 |- ( ph -> ( F oR U H <-> A. w e. A ( F ` w ) U ( H ` w ) ) )
43	25, 41, 42	3imtr4d 202	1 |- ( ph -> ( ( F oR R G /\ G oR T H ) -> F oR U H ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136   A.wral 2444   class class class wbr 3982   Fn wfn 5183   -->wf 5184   ` cfv 5188   oRcofr 6049

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ofr 6051

This theorem is referenced by: (None)