Theorem fmptdF 25994

Description: Domain and co-domain of the mapping operation; deduction form. This version of fmptd 5888 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 1706	. . . . 5 |- ( [ y / x ] ( ph /\ x e. A ) -> [ y / x ] B e. C )
3		sban 2091	. . . . . 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 2071	. . . . . . 7 |- ( [ y / x ] ph <-> ph )
6		fmptdF.a	. . . . . . . 8 |- F/_ x A
7	6	clelsb3f 25886	. . . . . . 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 3211	. . . . . 6 |- ( [ y / x ] B e. C <-> [. y / x ]. B e. C )
11		sbcel12 3696	. . . . . . 7 |- ( [. y / x ]. B e. C <-> [_ y / x ]_ B e. [_ y / x ]_ C )
12		vex 2996	. . . . . . . . 9 |- y e. _V
13		fmptdF.c	. . . . . . . . . 10 |- F/_ x C
14	13	csbconstgf 3321	. . . . . . . . 9 |- ( y e. _V -> [_ y / x ]_ C = C )
15	12, 14	ax-mp 5	. . . . . . . 8 |- [_ y / x ]_ C = C
16	15	eleq2i 2507	. . . . . . 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 2820	. . 3 |- ( ph -> A. y e. A [_ y / x ]_ B e. C )
21		nfcv 2589	. . . . 5 |- F/_ y A
22		nfcv 2589	. . . . 5 |- F/_ y B
23		nfcsb1v 3325	. . . . 5 |- F/_ x [_ y / x ]_ B
24		csbeq1a 3318	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
25	6, 21, 22, 23, 24	cbvmptf 25993	. . . 4 |- ( x e. A |-> B ) = ( y e. A |-> [_ y / x ]_ B )
26	25	fmpt 5885	. . 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 5572	. 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 1369   F/wnf 1589   [wsb 1700   e. wcel 1756   F/_wnfc 2575   A.wral 2736   _Vcvv 2993   [.wsbc 3207   [_csb 3309   e. cmpt 4371   -->wf 5435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447

This theorem is referenced by:  fmptcof2  26001  esumcl  26508  esumid  26521  esumval  26522  esumel  26523  esumsplit  26528  gsumesum  26532  esumlub  26533  esumaddf  26534  esumss  26543  esumpfinvalf  26547  voliune  26667