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Theorem fvss 5524
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 5219 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 iotass 5190 . 2 (∀𝑥(𝐴𝐹𝑥𝑥𝐵) → (℩𝑥𝐴𝐹𝑥) ⊆ 𝐵)
31, 2eqsstrid 3201 1 (∀𝑥(𝐴𝐹𝑥𝑥𝐵) → (𝐹𝐴) ⊆ 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1351  wss 3129   class class class wbr 4000  cio 5171  cfv 5211
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-uni 3808  df-iota 5173  df-fv 5219
This theorem is referenced by:  fvssunirng  5525  relfvssunirn  5526  sefvex  5531  fvmptss2  5586  tfrexlem  6328
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