Theorem fmptdF 28267

Description: Domain and co-domain of the mapping operation; deduction form. This version of fmptd 6051 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 1805	. . . . 5 |- ( [ y / x ] ( ph /\ x e. A ) -> [ y / x ] B e. C )
3		sban 2230	. . . . . 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 2211	. . . . . . 7 |- ( [ y / x ] ph <-> ph )
6		fmptdF.a	. . . . . . . 8 |- F/_ x A
7	6	clelsb3f 28128	. . . . . . 7 |- ( [ y / x ] x e. A <-> y e. A )
8	5, 7	anbi12i 704	. . . . . 6 |- ( ( [ y / x ] ph /\ [ y / x ] x e. A ) <-> ( ph /\ y e. A ) )
9	3, 8	bitri 253	. . . . 5 |- ( [ y / x ] ( ph /\ x e. A ) <-> ( ph /\ y e. A ) )
10		sbsbc 3273	. . . . . 6 |- ( [ y / x ] B e. C <-> [. y / x ]. B e. C )
11		sbcel12 3774	. . . . . . 7 |- ( [. y / x ]. B e. C <-> [_ y / x ]_ B e. [_ y / x ]_ C )
12		vex 3050	. . . . . . . . 9 |- y e. _V
13		fmptdF.c	. . . . . . . . . 10 |- F/_ x C
14	13	csbconstgf 3377	. . . . . . . . 9 |- ( y e. _V -> [_ y / x ]_ C = C )
15	12, 14	ax-mp 5	. . . . . . . 8 |- [_ y / x ]_ C = C
16	15	eleq2i 2523	. . . . . . 7 |- ( [_ y / x ]_ B e. [_ y / x ]_ C <-> [_ y / x ]_ B e. C )
17	11, 16	bitri 253	. . . . . 6 |- ( [. y / x ]. B e. C <-> [_ y / x ]_ B e. C )
18	10, 17	bitri 253	. . . . 5 |- ( [ y / x ] B e. C <-> [_ y / x ]_ B e. C )
19	2, 9, 18	3imtr3i 269	. . . 4 |- ( ( ph /\ y e. A ) -> [_ y / x ]_ B e. C )
20	19	ralrimiva 2804	. . 3 |- ( ph -> A. y e. A [_ y / x ]_ B e. C )
21		nfcv 2594	. . . . 5 |- F/_ y A
22		nfcv 2594	. . . . 5 |- F/_ y B
23		nfcsb1v 3381	. . . . 5 |- F/_ x [_ y / x ]_ B
24		csbeq1a 3374	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
25	6, 21, 22, 23, 24	cbvmptf 4496	. . . 4 |- ( x e. A |-> B ) = ( y e. A |-> [_ y / x ]_ B )
26	25	fmpt 6048	. . 3 |- ( A. y e. A [_ y / x ]_ B e. C <-> ( x e. A |-> B ) : A --> C )
27	20, 26	sylib 200	. 2 |- ( ph -> ( x e. A |-> B ) : A --> C )
28		fmptdF.2	. . 3 |- F = ( x e. A |-> B )
29	28	feq1i 5725	. 2 |- ( F : A --> C <-> ( x e. A |-> B ) : A --> C )
30	27, 29	sylibr 216	1 |- ( ph -> F : A --> C )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1446   F/wnf 1669   [wsb 1799   e. wcel 1889   F/_wnfc 2581   A.wral 2739   _Vcvv 3047   [.wsbc 3269   [_csb 3365   |-> cmpt 4464   -->wf 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-fv 5593

This theorem is referenced by:  fmptcof2  28271  esumcl  28863  esumid  28877  esumgsum  28878  esumval  28879  esumel  28880  esumsplit  28886  esumaddf  28894  esumss  28905  esumpfinvalf  28909