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Theorem fvss 5303
Description: The value of a function is a subset of 𝐵 if every element that could be a candidate for the value is a subset of 𝐵. (Contributed by Mario Carneiro, 24-May-2019.)
Assertion
Ref Expression
fvss (∀𝑥(𝐴𝐹𝑥𝑥𝐵) → (𝐹𝐴) ⊆ 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem fvss
StepHypRef Expression
1 df-fv 5010 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 iotass 4984 . 2 (∀𝑥(𝐴𝐹𝑥𝑥𝐵) → (℩𝑥𝐴𝐹𝑥) ⊆ 𝐵)
31, 2syl5eqss 3068 1 (∀𝑥(𝐴𝐹𝑥𝑥𝐵) → (𝐹𝐴) ⊆ 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  wss 2997   class class class wbr 3837  cio 4965  cfv 5002
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-iota 4967  df-fv 5010
This theorem is referenced by:  fvssunirng  5304  relfvssunirn  5305  sefvex  5310  fvmptss2  5363  tfrexlem  6081
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