Theorem offveqb 6201

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 5647	. . . 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 3390	. . . 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 6189	. . 3 |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )
11	3, 10	eqeq12d 2222	. 2 |- ( ph -> ( H = ( F oF R G ) <-> ( x e. A |-> ( H ` x ) ) = ( x e. A |-> ( B R C ) ) ) )
12		funfvex 5616	. . . . . 6 |- ( ( Fun H /\ x e. dom H ) -> ( H ` x ) e. _V )
13	12	funfni 5395	. . . . 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 2581	. . 3 |- ( ph -> A. x e. A ( H ` x ) e. _V )
16		mpteqb 5693	. . 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   A.wral 2486   _Vcvv 2776   |-> cmpt 4121   Fn wfn 5285   ` cfv 5290  (class class class)co 5967   oFcof 6179

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-of 6181

This theorem is referenced by:  offveq  6202