Theorem offeq 6258

Description: Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)

Hypotheses

Ref	Expression
off.1	|- ( ( ph /\ ( x e. S /\ y e. T ) ) -> ( x R y ) e. U )
off.2	|- ( ph -> F : A --> S )
off.3	|- ( ph -> G : B --> T )
off.4	|- ( ph -> A e. V )
off.5	|- ( ph -> B e. W )
off.6	|- ( A i^i B ) = C
offeq.4	|- ( ph -> H : C --> U )
offeq.5	|- ( ( ph /\ x e. A ) -> ( F ` x ) = D )
offeq.6	|- ( ( ph /\ x e. B ) -> ( G ` x ) = E )
offeq.7	|- ( ( ph /\ x e. C ) -> ( D R E ) = ( H ` x ) )

Assertion

Ref	Expression
offeq	|- ( ph -> ( F oF R G ) = H )

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

Allowed substitution hints:   A( x, y)   B( x, y)   C( y)   D( x, y)   E( x, y)   H( y)   V( x, y)   W( x, y)

Proof of Theorem offeq

Step	Hyp	Ref	Expression
1		off.1	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. T ) ) -> ( x R y ) e. U )
2		off.2	. . . 4 |- ( ph -> F : A --> S )
3		off.3	. . . 4 |- ( ph -> G : B --> T )
4		off.4	. . . 4 |- ( ph -> A e. V )
5		off.5	. . . 4 |- ( ph -> B e. W )
6		off.6	. . . 4 |- ( A i^i B ) = C
7	1, 2, 3, 4, 5, 6	off 6257	. . 3 |- ( ph -> ( F oF R G ) : C --> U )
8	7	ffnd 5490	. 2 |- ( ph -> ( F oF R G ) Fn C )
9		offeq.4	. . 3 |- ( ph -> H : C --> U )
10	9	ffnd 5490	. 2 |- ( ph -> H Fn C )
11	2	ffnd 5490	. . . 4 |- ( ph -> F Fn A )
12	3	ffnd 5490	. . . 4 |- ( ph -> G Fn B )
13		offeq.5	. . . 4 |- ( ( ph /\ x e. A ) -> ( F ` x ) = D )
14		offeq.6	. . . 4 |- ( ( ph /\ x e. B ) -> ( G ` x ) = E )
15		offeq.7	. . . . 5 |- ( ( ph /\ x e. C ) -> ( D R E ) = ( H ` x ) )
16	9	ffvelcdmda 5790	. . . . 5 |- ( ( ph /\ x e. C ) -> ( H ` x ) e. U )
17	15, 16	eqeltrd 2308	. . . 4 |- ( ( ph /\ x e. C ) -> ( D R E ) e. U )
18	11, 12, 4, 5, 6, 13, 14, 17	ofvalg 6254	. . 3 |- ( ( ph /\ x e. C ) -> ( ( F oF R G ) ` x ) = ( D R E ) )
19	18, 15	eqtrd 2264	. 2 |- ( ( ph /\ x e. C ) -> ( ( F oF R G ) ` x ) = ( H ` x ) )
20	8, 10, 19	eqfnfvd 5756	1 |- ( ph -> ( F oF R G ) = H )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   i^i cin 3200   -->wf 5329   ` cfv 5333  (class class class)co 6028   oFcof 6242

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-of 6244

This theorem is referenced by:  ofnegsub  9201  dviaddf  15516