Theorem caoftrn 6932

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 2972	. . . . 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 481	. . . 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 6359	. . . . 5 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
6		caofcom.3	. . . . . 6 |- ( ph -> G : A --> S )
7	6	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ w e. A ) -> ( G ` w ) e. S )
8		caofass.4	. . . . . 6 |- ( ph -> H : A --> S )
9	8	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ w e. A ) -> ( H ` w ) e. S )
10		breq1 4656	. . . . . . . 8 |- ( x = ( F ` w ) -> ( x R y <-> ( F ` w ) R y ) )
11	10	anbi1d 741	. . . . . . 7 |- ( x = ( F ` w ) -> ( ( x R y /\ y T z ) <-> ( ( F ` w ) R y /\ y T z ) ) )
12		breq1 4656	. . . . . . 7 |- ( x = ( F ` w ) -> ( x U z <-> ( F ` w ) U z ) )
13	11, 12	imbi12d 334	. . . . . 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 4657	. . . . . . . 8 |- ( y = ( G ` w ) -> ( ( F ` w ) R y <-> ( F ` w ) R ( G ` w ) ) )
15		breq1 4656	. . . . . . . 8 |- ( y = ( G ` w ) -> ( y T z <-> ( G ` w ) T z ) )
16	14, 15	anbi12d 747	. . . . . . 7 |- ( y = ( G ` w ) -> ( ( ( F ` w ) R y /\ y T z ) <-> ( ( F ` w ) R ( G ` w ) /\ ( G ` w ) T z ) ) )
17	16	imbi1d 331	. . . . . 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 4657	. . . . . . . 8 |- ( z = ( H ` w ) -> ( ( G ` w ) T z <-> ( G ` w ) T ( H ` w ) ) )
19	18	anbi2d 740	. . . . . . 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 4657	. . . . . . 7 |- ( z = ( H ` w ) -> ( ( F ` w ) U z <-> ( F ` w ) U ( H ` w ) ) )
21	19, 20	imbi12d 334	. . . . . 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 3325	. . . . 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 1326	. . . 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 15	. . 3 |- ( ( ph /\ w e. A ) -> ( ( ( F ` w ) R ( G ` w ) /\ ( G ` w ) T ( H ` w ) ) -> ( F ` w ) U ( H ` w ) ) )
25	24	ralimdva 2962	. 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 6045	. . . . . 6 |- ( F : A --> S -> F Fn A )
27	4, 26	syl 17	. . . . 5 |- ( ph -> F Fn A )
28		ffn 6045	. . . . . 6 |- ( G : A --> S -> G Fn A )
29	6, 28	syl 17	. . . . 5 |- ( ph -> G Fn A )
30		caofref.1	. . . . 5 |- ( ph -> A e. V )
31		inidm 3822	. . . . 5 |- ( A i^i A ) = A
32		eqidd 2623	. . . . 5 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
33		eqidd 2623	. . . . 5 |- ( ( ph /\ w e. A ) -> ( G ` w ) = ( G ` w ) )
34	27, 29, 30, 30, 31, 32, 33	ofrfval 6905	. . . 4 |- ( ph -> ( F oR R G <-> A. w e. A ( F ` w ) R ( G ` w ) ) )
35		ffn 6045	. . . . . 6 |- ( H : A --> S -> H Fn A )
36	8, 35	syl 17	. . . . 5 |- ( ph -> H Fn A )
37		eqidd 2623	. . . . 5 |- ( ( ph /\ w e. A ) -> ( H ` w ) = ( H ` w ) )
38	29, 36, 30, 30, 31, 33, 37	ofrfval 6905	. . . 4 |- ( ph -> ( G oR T H <-> A. w e. A ( G ` w ) T ( H ` w ) ) )
39	34, 38	anbi12d 747	. . 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 3064	. . 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 278	. 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 6905	. 2 |- ( ph -> ( F oR U H <-> A. w e. A ( F ` w ) U ( H ` w ) ) )
43	25, 41, 42	3imtr4d 283	1 |- ( ph -> ( ( F oR R G /\ G oR T H ) -> F oR U H ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   Fn wfn 5883   -->wf 5884   ` cfv 5888   oRcofr 6896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ofr 6898

This theorem is referenced by:  gsumbagdiaglem  19375  itg2le  23506