Theorem rnmptss 5657

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 5646	. 2 |- ( A. x e. A B e. C <-> F : A --> C )
3		frn 5356	. 2 |- ( F : A --> C -> ran F C_ C )
4	2, 3	sylbi 120	1 |- ( A. x e. A B e. C -> ran F C_ C )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   A.wral 2448   C_ wss 3121   |-> cmpt 4050   ran crn 4612   -->wf 5194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206

This theorem is referenced by:  mptexw  6092  tgiun  12867  dvrecap  13471