Theorem offeq 6248

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 6247	. . 3 |- ( ph -> ( F oF R G ) : C --> U )
8	7	ffnd 5483	. 2 |- ( ph -> ( F oF R G ) Fn C )
9		offeq.4	. . 3 |- ( ph -> H : C --> U )
10	9	ffnd 5483	. 2 |- ( ph -> H Fn C )
11	2	ffnd 5483	. . . 4 |- ( ph -> F Fn A )
12	3	ffnd 5483	. . . 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 5782	. . . . 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 6244	. . 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 5747	1 |- ( ph -> ( F oF R G ) = H )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   i^i cin 3199   -->wf 5322   ` cfv 5326  (class class class)co 6017   oFcof 6232

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-of 6234

This theorem is referenced by:  ofnegsub  9141  dviaddf  15428