Theorem fmptdF 29456

Description: Domain and co-domain of the mapping operation; deduction form. This version of fmptd 6385 uses bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.)

Hypotheses

Ref	Expression
fmptdF.p	|- F/ x ph
fmptdF.a	|- F/_ x A
fmptdF.c	|- F/_ x C
fmptdF.1	|- ( ( ph /\ x e. A ) -> B e. C )
fmptdF.2	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
fmptdF	|- ( ph -> F : A --> C )

Proof of Theorem fmptdF

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmptdF.1	. . . . . 6 |- ( ( ph /\ x e. A ) -> B e. C )
2	1	sbimi 1886	. . . . 5 |- ( [ y / x ] ( ph /\ x e. A ) -> [ y / x ] B e. C )
3		sban 2399	. . . . . 6 |- ( [ y / x ] ( ph /\ x e. A ) <-> ( [ y / x ] ph /\ [ y / x ] x e. A ) )
4		fmptdF.p	. . . . . . . 8 |- F/ x ph
5	4	sbf 2380	. . . . . . 7 |- ( [ y / x ] ph <-> ph )
6		fmptdF.a	. . . . . . . 8 |- F/_ x A
7	6	clelsb3f 2768	. . . . . . 7 |- ( [ y / x ] x e. A <-> y e. A )
8	5, 7	anbi12i 733	. . . . . 6 |- ( ( [ y / x ] ph /\ [ y / x ] x e. A ) <-> ( ph /\ y e. A ) )
9	3, 8	bitri 264	. . . . 5 |- ( [ y / x ] ( ph /\ x e. A ) <-> ( ph /\ y e. A ) )
10		sbsbc 3439	. . . . . 6 |- ( [ y / x ] B e. C <-> [. y / x ]. B e. C )
11		sbcel12 3983	. . . . . . 7 |- ( [. y / x ]. B e. C <-> [_ y / x ]_ B e. [_ y / x ]_ C )
12		vex 3203	. . . . . . . . 9 |- y e. _V
13		fmptdF.c	. . . . . . . . . 10 |- F/_ x C
14	13	csbconstgf 3545	. . . . . . . . 9 |- ( y e. _V -> [_ y / x ]_ C = C )
15	12, 14	ax-mp 5	. . . . . . . 8 |- [_ y / x ]_ C = C
16	15	eleq2i 2693	. . . . . . 7 |- ( [_ y / x ]_ B e. [_ y / x ]_ C <-> [_ y / x ]_ B e. C )
17	11, 16	bitri 264	. . . . . 6 |- ( [. y / x ]. B e. C <-> [_ y / x ]_ B e. C )
18	10, 17	bitri 264	. . . . 5 |- ( [ y / x ] B e. C <-> [_ y / x ]_ B e. C )
19	2, 9, 18	3imtr3i 280	. . . 4 |- ( ( ph /\ y e. A ) -> [_ y / x ]_ B e. C )
20	19	ralrimiva 2966	. . 3 |- ( ph -> A. y e. A [_ y / x ]_ B e. C )
21		nfcv 2764	. . . . 5 |- F/_ y A
22		nfcv 2764	. . . . 5 |- F/_ y B
23		nfcsb1v 3549	. . . . 5 |- F/_ x [_ y / x ]_ B
24		csbeq1a 3542	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
25	6, 21, 22, 23, 24	cbvmptf 4748	. . . 4 |- ( x e. A |-> B ) = ( y e. A |-> [_ y / x ]_ B )
26	25	fmpt 6381	. . 3 |- ( A. y e. A [_ y / x ]_ B e. C <-> ( x e. A |-> B ) : A --> C )
27	20, 26	sylib 208	. 2 |- ( ph -> ( x e. A |-> B ) : A --> C )
28		fmptdF.2	. . 3 |- F = ( x e. A |-> B )
29	28	feq1i 6036	. 2 |- ( F : A --> C <-> ( x e. A |-> B ) : A --> C )
30	27, 29	sylibr 224	1 |- ( ph -> F : A --> C )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   [wsb 1880   e. wcel 1990   F/_wnfc 2751   A.wral 2912   _Vcvv 3200   [.wsbc 3435   [_csb 3533   |-> cmpt 4729   -->wf 5884

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-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-fal 1489  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-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-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-fv 5896

This theorem is referenced by:  fmptcof2  29457  esumcl  30092  esumid  30106  esumgsum  30107  esumval  30108  esumel  30109  esumsplit  30115  esumaddf  30123  esumss  30134  esumpfinvalf  30138