Theorem offval2f 6909

Description: The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 23-Jun-2017.)

Hypotheses

Ref	Expression
offval2f.0	|- F/ x ph
offval2f.a	|- F/_ x A
offval2f.1	|- ( ph -> A e. V )
offval2f.2	|- ( ( ph /\ x e. A ) -> B e. W )
offval2f.3	|- ( ( ph /\ x e. A ) -> C e. X )
offval2f.4	|- ( ph -> F = ( x e. A |-> B ) )
offval2f.5	|- ( ph -> G = ( x e. A |-> C ) )

Assertion

Ref	Expression
offval2f	|- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )

Distinct variable group:   x, R

Allowed substitution hints:   ph( x)   A( x)   B( x)   C( x)   F( x)   G( x)   V( x)   W( x)   X( x)

Proof of Theorem offval2f

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval2f.0	. . . . . 6 |- F/ x ph
2		offval2f.2	. . . . . . 7 |- ( ( ph /\ x e. A ) -> B e. W )
3	2	ex 450	. . . . . 6 |- ( ph -> ( x e. A -> B e. W ) )
4	1, 3	ralrimi 2957	. . . . 5 |- ( ph -> A. x e. A B e. W )
5		offval2f.a	. . . . . 6 |- F/_ x A
6	5	fnmptf 6016	. . . . 5 |- ( A. x e. A B e. W -> ( x e. A |-> B ) Fn A )
7	4, 6	syl 17	. . . 4 |- ( ph -> ( x e. A |-> B ) Fn A )
8		offval2f.4	. . . . 5 |- ( ph -> F = ( x e. A |-> B ) )
9	8	fneq1d 5981	. . . 4 |- ( ph -> ( F Fn A <-> ( x e. A |-> B ) Fn A ) )
10	7, 9	mpbird 247	. . 3 |- ( ph -> F Fn A )
11		offval2f.3	. . . . . . 7 |- ( ( ph /\ x e. A ) -> C e. X )
12	11	ex 450	. . . . . 6 |- ( ph -> ( x e. A -> C e. X ) )
13	1, 12	ralrimi 2957	. . . . 5 |- ( ph -> A. x e. A C e. X )
14	5	fnmptf 6016	. . . . 5 |- ( A. x e. A C e. X -> ( x e. A |-> C ) Fn A )
15	13, 14	syl 17	. . . 4 |- ( ph -> ( x e. A |-> C ) Fn A )
16		offval2f.5	. . . . 5 |- ( ph -> G = ( x e. A |-> C ) )
17	16	fneq1d 5981	. . . 4 |- ( ph -> ( G Fn A <-> ( x e. A |-> C ) Fn A ) )
18	15, 17	mpbird 247	. . 3 |- ( ph -> G Fn A )
19		offval2f.1	. . 3 |- ( ph -> A e. V )
20		inidm 3822	. . 3 |- ( A i^i A ) = A
21	8	adantr 481	. . . 4 |- ( ( ph /\ y e. A ) -> F = ( x e. A |-> B ) )
22	21	fveq1d 6193	. . 3 |- ( ( ph /\ y e. A ) -> ( F ` y ) = ( ( x e. A |-> B ) ` y ) )
23	16	adantr 481	. . . 4 |- ( ( ph /\ y e. A ) -> G = ( x e. A |-> C ) )
24	23	fveq1d 6193	. . 3 |- ( ( ph /\ y e. A ) -> ( G ` y ) = ( ( x e. A |-> C ) ` y ) )
25	10, 18, 19, 19, 20, 22, 24	offval 6904	. 2 |- ( ph -> ( F oF R G ) = ( y e. A |-> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) ) )
26		nfcv 2764	. . . 4 |- F/_ y A
27		nffvmpt1 6199	. . . . 5 |- F/_ x ( ( x e. A |-> B ) ` y )
28		nfcv 2764	. . . . 5 |- F/_ x R
29		nffvmpt1 6199	. . . . 5 |- F/_ x ( ( x e. A |-> C ) ` y )
30	27, 28, 29	nfov 6676	. . . 4 |- F/_ x ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) )
31		nfcv 2764	. . . 4 |- F/_ y ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) )
32		fveq2 6191	. . . . 5 |- ( y = x -> ( ( x e. A |-> B ) ` y ) = ( ( x e. A |-> B ) ` x ) )
33		fveq2 6191	. . . . 5 |- ( y = x -> ( ( x e. A |-> C ) ` y ) = ( ( x e. A |-> C ) ` x ) )
34	32, 33	oveq12d 6668	. . . 4 |- ( y = x -> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) = ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) )
35	26, 5, 30, 31, 34	cbvmptf 4748	. . 3 |- ( y e. A |-> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) ) = ( x e. A |-> ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) )
36		simpr 477	. . . . . 6 |- ( ( ph /\ x e. A ) -> x e. A )
37	5	fvmpt2f 6283	. . . . . 6 |- ( ( x e. A /\ B e. W ) -> ( ( x e. A |-> B ) ` x ) = B )
38	36, 2, 37	syl2anc 693	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> B ) ` x ) = B )
39	5	fvmpt2f 6283	. . . . . 6 |- ( ( x e. A /\ C e. X ) -> ( ( x e. A |-> C ) ` x ) = C )
40	36, 11, 39	syl2anc 693	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> C ) ` x ) = C )
41	38, 40	oveq12d 6668	. . . 4 |- ( ( ph /\ x e. A ) -> ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) = ( B R C ) )
42	1, 41	mpteq2da 4743	. . 3 |- ( ph -> ( x e. A |-> ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) ) = ( x e. A |-> ( B R C ) ) )
43	35, 42	syl5eq 2668	. 2 |- ( ph -> ( y e. A |-> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) ) = ( x e. A |-> ( B R C ) ) )
44	25, 43	eqtrd 2656	1 |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   A.wral 2912   |-> cmpt 4729   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   oFcof 6895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897

This theorem is referenced by:  esumaddf  30123  binomcxplemnotnn0  38555