Theorem fmptdF 27643

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 1746	. . . . 5 |- ( [ y / x ] ( ph /\ x e. A ) -> [ y / x ] B e. C )
3		sban 2141	. . . . . 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 2122	. . . . . . 7 |- ( [ y / x ] ph <-> ph )
6		fmptdF.a	. . . . . . . 8 |- F/_ x A
7	6	clelsb3f 27506	. . . . . . 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 3331	. . . . . 6 |- ( [ y / x ] B e. C <-> [. y / x ]. B e. C )
11		sbcel12 3832	. . . . . . 7 |- ( [. y / x ]. B e. C <-> [_ y / x ]_ B e. [_ y / x ]_ C )
12		vex 3112	. . . . . . . . 9 |- y e. _V
13		fmptdF.c	. . . . . . . . . 10 |- F/_ x C
14	13	csbconstgf 3442	. . . . . . . . 9 |- ( y e. _V -> [_ y / x ]_ C = C )
15	12, 14	ax-mp 5	. . . . . . . 8 |- [_ y / x ]_ C = C
16	15	eleq2i 2535	. . . . . . 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 2871	. . 3 |- ( ph -> A. y e. A [_ y / x ]_ B e. C )
21		nfcv 2619	. . . . 5 |- F/_ y A
22		nfcv 2619	. . . . 5 |- F/_ y B
23		nfcsb1v 3446	. . . . 5 |- F/_ x [_ y / x ]_ B
24		csbeq1a 3439	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
25	6, 21, 22, 23, 24	cbvmptf 27642	. . . 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 1395   F/wnf 1617   [wsb 1740   e. wcel 1819   F/_wnfc 2605   A.wral 2807   _Vcvv 3109   [.wsbc 3327   [_csb 3430   |-> cmpt 4515   -->wf 5590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602

This theorem is referenced by:  fmptcof2  27651  esumcl  28204  esumid  28218  esumgsum  28219  esumval  28220  esumel  28221  esumsplit  28227  esumaddf  28233  esumss  28244  esumpfinvalf  28248