Theorem offeq 6316

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 6315	. . 3 |- ( ph -> ( F oF R G ) : C --> U )
8	7	ffnd 5534	. 2 |- ( ph -> ( F oF R G ) Fn C )
9		offeq.4	. . 3 |- ( ph -> H : C --> U )
10	9	ffnd 5534	. 2 |- ( ph -> H Fn C )
11	2	ffnd 5534	. . . 4 |- ( ph -> F Fn A )
12	3	ffnd 5534	. . . 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 5843	. . . . 5 |- ( ( ph /\ x e. C ) -> ( H ` x ) e. U )
17	15, 16	eqeltrd 2315	. . . 4 |- ( ( ph /\ x e. C ) -> ( D R E ) e. U )
18	11, 12, 4, 5, 6, 13, 14, 17	ofvalg 6312	. . 3 |- ( ( ph /\ x e. C ) -> ( ( F oF R G ) ` x ) = ( D R E ) )
19	18, 15	eqtrd 2271	. 2 |- ( ( ph /\ x e. C ) -> ( ( F oF R G ) ` x ) = ( H ` x ) )
20	8, 10, 19	eqfnfvd 5809	1 |- ( ph -> ( F oF R G ) = H )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   i^i cin 3219   -->wf 5373   ` cfv 5377  (class class class)co 6085   oFcof 6300

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302

This theorem is used by:  ofnegsub  9292  dviaddf  15806