Theorem offveqb 6001

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 5467	. . . 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 3285	. . . 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 5989	. . 3 |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )
11	3, 10	eqeq12d 2154	. 2 |- ( ph -> ( H = ( F oF R G ) <-> ( x e. A |-> ( H ` x ) ) = ( x e. A |-> ( B R C ) ) ) )
12		funfvex 5438	. . . . . 6 |- ( ( Fun H /\ x e. dom H ) -> ( H ` x ) e. _V )
13	12	funfni 5223	. . . . 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 2505	. . 3 |- ( ph -> A. x e. A ( H ` x ) e. _V )
16		mpteqb 5511	. . 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 1331   e. wcel 1480   A.wral 2416   _Vcvv 2686   |-> cmpt 3989   Fn wfn 5118   ` cfv 5123  (class class class)co 5774   oFcof 5980

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-of 5982

This theorem is referenced by: (None)