Theorem fmptdF 27318

Description: Domain and co-domain of the mapping operation; deduction form. This version of fmptd 6056 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 1717	. . . . 5 |- ( [ y / x ] ( ph /\ x e. A ) -> [ y / x ] B e. C )
3		sban 2114	. . . . . 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 2094	. . . . . . 7 |- ( [ y / x ] ph <-> ph )
6		fmptdF.a	. . . . . . . 8 |- F/_ x A
7	6	clelsb3f 27202	. . . . . . 7 |- ( [ y / x ] x e. A <-> y e. A )
8	5, 7	anbi12i 697	. . . . . 6 |- ( ( [ y / x ] ph /\ [ y / x ] x e. A ) <-> ( ph /\ y e. A ) )
9	3, 8	bitri 249	. . . . 5 |- ( [ y / x ] ( ph /\ x e. A ) <-> ( ph /\ y e. A ) )
10		sbsbc 3340	. . . . . 6 |- ( [ y / x ] B e. C <-> [. y / x ]. B e. C )
11		sbcel12 3828	. . . . . . 7 |- ( [. y / x ]. B e. C <-> [_ y / x ]_ B e. [_ y / x ]_ C )
12		vex 3121	. . . . . . . . 9 |- y e. _V
13		fmptdF.c	. . . . . . . . . 10 |- F/_ x C
14	13	csbconstgf 3452	. . . . . . . . 9 |- ( y e. _V -> [_ y / x ]_ C = C )
15	12, 14	ax-mp 5	. . . . . . . 8 |- [_ y / x ]_ C = C
16	15	eleq2i 2545	. . . . . . 7 |- ( [_ y / x ]_ B e. [_ y / x ]_ C <-> [_ y / x ]_ B e. C )
17	11, 16	bitri 249	. . . . . 6 |- ( [. y / x ]. B e. C <-> [_ y / x ]_ B e. C )
18	10, 17	bitri 249	. . . . 5 |- ( [ y / x ] B e. C <-> [_ y / x ]_ B e. C )
19	2, 9, 18	3imtr3i 265	. . . 4 |- ( ( ph /\ y e. A ) -> [_ y / x ]_ B e. C )
20	19	ralrimiva 2881	. . 3 |- ( ph -> A. y e. A [_ y / x ]_ B e. C )
21		nfcv 2629	. . . . 5 |- F/_ y A
22		nfcv 2629	. . . . 5 |- F/_ y B
23		nfcsb1v 3456	. . . . 5 |- F/_ x [_ y / x ]_ B
24		csbeq1a 3449	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
25	6, 21, 22, 23, 24	cbvmptf 27317	. . . 4 |- ( x e. A |-> B ) = ( y e. A |-> [_ y / x ]_ B )
26	25	fmpt 6053	. . 3 |- ( A. y e. A [_ y / x ]_ B e. C <-> ( x e. A |-> B ) : A --> C )
27	20, 26	sylib 196	. 2 |- ( ph -> ( x e. A |-> B ) : A --> C )
28		fmptdF.2	. . 3 |- F = ( x e. A |-> B )
29	28	feq1i 5729	. 2 |- ( F : A --> C <-> ( x e. A |-> B ) : A --> C )
30	27, 29	sylibr 212	1 |- ( ph -> F : A --> C )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   F/wnf 1599   [wsb 1711   e. wcel 1767   F/_wnfc 2615   A.wral 2817   _Vcvv 3118   [.wsbc 3336   [_csb 3440   |-> cmpt 4511   -->wf 5590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602

This theorem is referenced by:  fmptcof2  27325  esumcl  27868  esumid  27881  esumval  27882  esumel  27883  esumsplit  27888  esumaddf  27894  esumss  27903  esumpfinvalf  27907  voliune  28026