Theorem offveqb 6069

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 5532	. . . 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 3331	. . . 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 6057	. . 3 |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )
11	3, 10	eqeq12d 2180	. 2 |- ( ph -> ( H = ( F oF R G ) <-> ( x e. A |-> ( H ` x ) ) = ( x e. A |-> ( B R C ) ) ) )
12		funfvex 5503	. . . . . 6 |- ( ( Fun H /\ x e. dom H ) -> ( H ` x ) e. _V )
13	12	funfni 5288	. . . . 5 |- ( ( H Fn A /\ x e. A ) -> ( H ` x ) e. _V )
14	1, 13	sylan 281	. . . 4 |- ( ( ph /\ x e. A ) -> ( H ` x ) e. _V )
15	14	ralrimiva 2539	. . 3 |- ( ph -> A. x e. A ( H ` x ) e. _V )
16		mpteqb 5576	. . 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 187	1 |- ( ph -> ( H = ( F oF R G ) <-> A. x e. A ( H ` x ) = ( B R C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   _Vcvv 2726   |-> cmpt 4043   Fn wfn 5183   ` cfv 5188  (class class class)co 5842   oFcof 6048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050

This theorem is referenced by: (None)