Theorem fvmptss2 5774

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.) (Revised by Mario Carneiro, 3-Jul-2019.)

Hypotheses

Ref	Expression
fvmptss2.1	|- ( x = D -> B = C )
fvmptss2.2	|- 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

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvss 5704	. 2 |- ( A. y ( D F y -> y C_ C ) -> ( F ` D ) C_ C )
2		fvmptss2.2	. . . . . 6 |- F = ( x e. A |-> B )
3	2	funmpt2 5411	. . . . 5 |- Fun F
4		funrel 5389	. . . . 5 |- ( Fun F -> Rel F )
5	3, 4	ax-mp 5	. . . 4 |- Rel F
6	5	brrelex1i 4813	. . 3 |- ( D F y -> D e. _V )
7		nfcv 2392	. . . 4 |- F/_ x D
8		nfmpt1 4219	. . . . . . 7 |- F/_ x ( x e. A |-> B )
9	2, 8	nfcxfr 2389	. . . . . 6 |- F/_ x F
10		nfcv 2392	. . . . . 6 |- F/_ x y
11	7, 9, 10	nfbr 4172	. . . . 5 |- F/ x D F y
12		nfv 1581	. . . . 5 |- F/ x y C_ C
13	11, 12	nfim 1625	. . . 4 |- F/ x ( D F y -> y C_ C )
14		breq1 4128	. . . . 5 |- ( x = D -> ( x F y <-> D F y ) )
15		fvmptss2.1	. . . . . 6 |- ( x = D -> B = C )
16	15	sseq2d 3278	. . . . 5 |- ( x = D -> ( y C_ B <-> y C_ C ) )
17	14, 16	imbi12d 234	. . . 4 |- ( x = D -> ( ( x F y -> y C_ B ) <-> ( D F y -> y C_ C ) ) )
18		df-br 4126	. . . . 5 |- ( x F y <-> <. x , y >. e. F )
19		opabid 4393	. . . . . . 7 |- ( <. x , y >. e. { <. x , y >. | ( x e. A /\ y = B ) } <-> ( x e. A /\ y = B ) )
20		eqimss 3302	. . . . . . . 8 |- ( y = B -> y C_ B )
21	20	adantl 277	. . . . . . 7 |- ( ( x e. A /\ y = B ) -> y C_ B )
22	19, 21	sylbi 121	. . . . . 6 |- ( <. x , y >. e. { <. x , y >. | ( x e. A /\ y = B ) } -> y C_ B )
23		df-mpt 4189	. . . . . . 7 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
24	2, 23	eqtri 2259	. . . . . 6 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
25	22, 24	eleq2s 2333	. . . . 5 |- ( <. x , y >. e. F -> y C_ B )
26	18, 25	sylbi 121	. . . 4 |- ( x F y -> y C_ B )
27	7, 13, 17, 26	vtoclgf 2881	. . 3 |- ( D e. _V -> ( D F y -> y C_ C ) )
28	6, 27	mpcom 36	. 2 |- ( D F y -> y C_ C )
29	1, 28	mpg 1504	1 |- ( F ` D ) C_ C

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   <.cop 3708   class class class wbr 4125   {copab 4186   |-> cmpt 4187   Rel wrel 4774   Fun wfun 5366   ` cfv 5372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-iota 5332  df-fun 5374  df-fv 5380

This theorem is referenced by:  mptfvex  5785