Theorem caoftrn 6007

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 2515	. . . . 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 5555	. . . . 5 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
6		caofcom.3	. . . . . 6 |- ( ph -> G : A --> S )
7	6	ffvelrnda 5555	. . . . 5 |- ( ( ph /\ w e. A ) -> ( G ` w ) e. S )
8		caofass.4	. . . . . 6 |- ( ph -> H : A --> S )
9	8	ffvelrnda 5555	. . . . 5 |- ( ( ph /\ w e. A ) -> ( H ` w ) e. S )
10		breq1 3932	. . . . . . . 8 |- ( x = ( F ` w ) -> ( x R y <-> ( F ` w ) R y ) )
11	10	anbi1d 460	. . . . . . 7 |- ( x = ( F ` w ) -> ( ( x R y /\ y T z ) <-> ( ( F ` w ) R y /\ y T z ) ) )
12		breq1 3932	. . . . . . 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 3933	. . . . . . . 8 |- ( y = ( G ` w ) -> ( ( F ` w ) R y <-> ( F ` w ) R ( G ` w ) ) )
15		breq1 3932	. . . . . . . 8 |- ( y = ( G ` w ) -> ( y T z <-> ( G ` w ) T z ) )
16	14, 15	anbi12d 464	. . . . . . 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 3933	. . . . . . . 8 |- ( z = ( H ` w ) -> ( ( G ` w ) T z <-> ( G ` w ) T ( H ` w ) ) )
19	18	anbi2d 459	. . . . . . 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 3933	. . . . . . 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 2805	. . . . 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 1216	. . . 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 2499	. 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 5272	. . . . . 6 |- ( F : A --> S -> F Fn A )
27	4, 26	syl 14	. . . . 5 |- ( ph -> F Fn A )
28		ffn 5272	. . . . . 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 3285	. . . . 5 |- ( A i^i A ) = A
32		eqidd 2140	. . . . 5 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
33		eqidd 2140	. . . . 5 |- ( ( ph /\ w e. A ) -> ( G ` w ) = ( G ` w ) )
34	27, 29, 30, 30, 31, 32, 33	ofrfval 5990	. . . 4 |- ( ph -> ( F oR R G <-> A. w e. A ( F ` w ) R ( G ` w ) ) )
35		ffn 5272	. . . . . 6 |- ( H : A --> S -> H Fn A )
36	8, 35	syl 14	. . . . 5 |- ( ph -> H Fn A )
37		eqidd 2140	. . . . 5 |- ( ( ph /\ w e. A ) -> ( H ` w ) = ( H ` w ) )
38	29, 36, 30, 30, 31, 33, 37	ofrfval 5990	. . . 4 |- ( ph -> ( G oR T H <-> A. w e. A ( G ` w ) T ( H ` w ) ) )
39	34, 38	anbi12d 464	. . 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 2558	. . 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	syl6bbr 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 5990	. 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 962   = 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)