Theorem offveqb 6322

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 5748	. . . 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 3440	. . . 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 6310	. . 3 |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )
11	3, 10	eqeq12d 2253	. 2 |- ( ph -> ( H = ( F oF R G ) <-> ( x e. A |-> ( H ` x ) ) = ( x e. A |-> ( B R C ) ) ) )
12		funfvex 5712	. . . . . 6 |- ( ( Fun H /\ x e. dom H ) -> ( H ` x ) e. _V )
13	12	funfni 5483	. . . . 5 |- ( ( H Fn A /\ x e. A ) -> ( H ` x ) e. _V )
14	1, 13	sylan 283	. . . 4 |- ( ( ph /\ x e. A ) -> ( H ` x ) e. _V )
15	14	ralrimiva 2623	. . 3 |- ( ph -> A. x e. A ( H ` x ) e. _V )
16		mpteqb 5796	. . 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 188	1 |- ( ph -> ( H = ( F oF R G ) <-> A. x e. A ( H ` x ) = ( B R C ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   |-> cmpt 4192   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   oFcof 6300

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302

This theorem is used by:  offveq  6323