Theorem fvmptss2 6304

Description: A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.)

Hypotheses

Ref	Expression
fvmptn.1	|- ( x = D -> B = C )
fvmptn.2	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
fvmptss2	|- ( F ` D ) C_ C

Distinct variable groups:   x, A   x, C   x, D

Allowed substitution hints:   B( x)   F( x)

Proof of Theorem fvmptss2

Step	Hyp	Ref	Expression
1		fvmptn.1	. . . . 5 |- ( x = D -> B = C )
2	1	eleq1d 2686	. . . 4 |- ( x = D -> ( B e. _V <-> C e. _V ) )
3		fvmptn.2	. . . . 5 |- F = ( x e. A |-> B )
4	3	dmmpt 5630	. . . 4 |- dom F = { x e. A | B e. _V }
5	2, 4	elrab2 3366	. . 3 |- ( D e. dom F <-> ( D e. A /\ C e. _V ) )
6	1, 3	fvmptg 6280	. . . 4 |- ( ( D e. A /\ C e. _V ) -> ( F ` D ) = C )
7		eqimss 3657	. . . 4 |- ( ( F ` D ) = C -> ( F ` D ) C_ C )
8	6, 7	syl 17	. . 3 |- ( ( D e. A /\ C e. _V ) -> ( F ` D ) C_ C )
9	5, 8	sylbi 207	. 2 |- ( D e. dom F -> ( F ` D ) C_ C )
10		ndmfv 6218	. . 3 |- ( -. D e. dom F -> ( F ` D ) = (/) )
11		0ss 3972	. . 3 |- (/) C_ C
12	10, 11	syl6eqss 3655	. 2 |- ( -. D e. dom F -> ( F ` D ) C_ C )
13	9, 12	pm2.61i 176	1 |- ( F ` D ) C_ C

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   dom cdm 5114   ` cfv 5888

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-8 1992  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-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-fv 5896

This theorem is referenced by:  cvmsi  31247