Theorem rnmptss 5347

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 5340	. 2 |- ( A. x e. A B e. C <-> F : A --> C )
3		frn 5072	. 2 |- ( F : A --> C -> ran F C_ C )
4	2, 3	sylbi 119	1 |- ( A. x e. A B e. C -> ran F C_ C )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   A.wral 2348   C_ wss 2973   |-> cmpt 3839   ran crn 4364   -->wf 4918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930

This theorem is referenced by: (None)