Theorem offeq 6149

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 6148	. . 3 |- ( ph -> ( F oF R G ) : C --> U )
8	7	ffnd 5408	. 2 |- ( ph -> ( F oF R G ) Fn C )
9		offeq.4	. . 3 |- ( ph -> H : C --> U )
10	9	ffnd 5408	. 2 |- ( ph -> H Fn C )
11	2	ffnd 5408	. . . 4 |- ( ph -> F Fn A )
12	3	ffnd 5408	. . . 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 5697	. . . . 5 |- ( ( ph /\ x e. C ) -> ( H ` x ) e. U )
17	15, 16	eqeltrd 2273	. . . 4 |- ( ( ph /\ x e. C ) -> ( D R E ) e. U )
18	11, 12, 4, 5, 6, 13, 14, 17	ofvalg 6145	. . 3 |- ( ( ph /\ x e. C ) -> ( ( F oF R G ) ` x ) = ( D R E ) )
19	18, 15	eqtrd 2229	. 2 |- ( ( ph /\ x e. C ) -> ( ( F oF R G ) ` x ) = ( H ` x ) )
20	8, 10, 19	eqfnfvd 5662	1 |- ( ph -> ( F oF R G ) = H )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   i^i cin 3156   -->wf 5254   ` cfv 5258  (class class class)co 5922   oFcof 6133

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-of 6135

This theorem is referenced by:  ofnegsub  8989  dviaddf  14941