Theorem offveqb 5761

Description: Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Hypotheses

Ref	Expression
offveq.1	|- ( ph -> A e. V )
offveq.2	|- ( ph -> F Fn A )
offveq.3	|- ( ph -> G Fn A )
offveq.4	|- ( ph -> H Fn A )
offveq.5	|- ( ( ph /\ x e. A ) -> ( F ` x ) = B )
offveq.6	|- ( ( ph /\ x e. A ) -> ( G ` x ) = C )

Assertion

Ref	Expression
offveqb	|- ( ph -> ( H = ( F oF R G ) <-> A. x e. A ( H ` x ) = ( B R C ) ) )

Distinct variable groups:   x, A   x, F   x, G   x, H   ph, x   x, R

Allowed substitution hints:   B( x)   C( x)   V( x)

Proof of Theorem offveqb

Step	Hyp	Ref	Expression
1		offveq.4	. . . 4 |- ( ph -> H Fn A )
2		dffn5im 5251	. . . 4 |- ( H Fn A -> H = ( x e. A |-> ( H ` x ) ) )
3	1, 2	syl 14	. . 3 |- ( ph -> H = ( x e. A |-> ( H ` x ) ) )
4		offveq.2	. . . 4 |- ( ph -> F Fn A )
5		offveq.3	. . . 4 |- ( ph -> G Fn A )
6		offveq.1	. . . 4 |- ( ph -> A e. V )
7		inidm 3182	. . . 4 |- ( A i^i A ) = A
8		offveq.5	. . . 4 |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )
9		offveq.6	. . . 4 |- ( ( ph /\ x e. A ) -> ( G ` x ) = C )
10	4, 5, 6, 6, 7, 8, 9	offval 5750	. . 3 |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )
11	3, 10	eqeq12d 2096	. 2 |- ( ph -> ( H = ( F oF R G ) <-> ( x e. A |-> ( H ` x ) ) = ( x e. A |-> ( B R C ) ) ) )
12		funfvex 5223	. . . . . 6 |- ( ( Fun H /\ x e. dom H ) -> ( H ` x ) e. _V )
13	12	funfni 5030	. . . . 5 |- ( ( H Fn A /\ x e. A ) -> ( H ` x ) e. _V )
14	1, 13	sylan 277	. . . 4 |- ( ( ph /\ x e. A ) -> ( H ` x ) e. _V )
15	14	ralrimiva 2435	. . 3 |- ( ph -> A. x e. A ( H ` x ) e. _V )
16		mpteqb 5293	. . 3 |- ( A. x e. A ( H ` x ) e. _V -> ( ( x e. A |-> ( H ` x ) ) = ( x e. A |-> ( B R C ) ) <-> A. x e. A ( H ` x ) = ( B R C ) ) )
17	15, 16	syl 14	. 2 |- ( ph -> ( ( x e. A |-> ( H ` x ) ) = ( x e. A |-> ( B R C ) ) <-> A. x e. A ( H ` x ) = ( B R C ) ) )
18	11, 17	bitrd 186	1 |- ( ph -> ( H = ( F oF R G ) <-> A. x e. A ( H ` x ) = ( B R C ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   e. wcel 1434   A.wral 2349   _Vcvv 2602   |-> cmpt 3847   Fn wfn 4927   ` cfv 4932  (class class class)co 5543   oFcof 5741

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-of 5743

This theorem is referenced by: (None)