Theorem offeq 6146

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 6145	. . 3 |- ( ph -> ( F oF R G ) : C --> U )
8	7	ffnd 5405	. 2 |- ( ph -> ( F oF R G ) Fn C )
9		offeq.4	. . 3 |- ( ph -> H : C --> U )
10	9	ffnd 5405	. 2 |- ( ph -> H Fn C )
11	2	ffnd 5405	. . . 4 |- ( ph -> F Fn A )
12	3	ffnd 5405	. . . 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 5694	. . . . 5 |- ( ( ph /\ x e. C ) -> ( H ` x ) e. U )
17	15, 16	eqeltrd 2270	. . . 4 |- ( ( ph /\ x e. C ) -> ( D R E ) e. U )
18	11, 12, 4, 5, 6, 13, 14, 17	ofvalg 6142	. . 3 |- ( ( ph /\ x e. C ) -> ( ( F oF R G ) ` x ) = ( D R E ) )
19	18, 15	eqtrd 2226	. 2 |- ( ( ph /\ x e. C ) -> ( ( F oF R G ) ` x ) = ( H ` x ) )
20	8, 10, 19	eqfnfvd 5659	1 |- ( ph -> ( F oF R G ) = H )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   i^i cin 3153   -->wf 5251   ` cfv 5255  (class class class)co 5919   oFcof 6130

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-of 6132

This theorem is referenced by:  ofnegsub  8983  dviaddf  14884