Theorem rnmptss 5796

Description: The range of an operation given by the maps-to notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)

Hypothesis

Ref	Expression
rnmptss.1	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
rnmptss	|- ( A. x e. A B e. C -> ran F C_ C )

Distinct variable groups:   x, A   x, C

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

Proof of Theorem rnmptss

Step	Hyp	Ref	Expression
1		rnmptss.1	. . 3 |- F = ( x e. A |-> B )
2	1	fmpt 5785	. 2 |- ( A. x e. A B e. C <-> F : A --> C )
3		frn 5482	. 2 |- ( F : A --> C -> ran F C_ C )
4	2, 3	sylbi 121	1 |- ( A. x e. A B e. C -> ran F C_ C )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   |-> cmpt 4145   ran crn 4720   -->wf 5314

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326

This theorem is referenced by:  mptexw  6258  tgiun  14747  dvrecap  15387