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Theorem fvss 5217
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 4938 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 iotass 4912 . 2 (∀𝑥(𝐴𝐹𝑥𝑥𝐵) → (℩𝑥𝐴𝐹𝑥) ⊆ 𝐵)
31, 2syl5eqss 3017 1 (∀𝑥(𝐴𝐹𝑥𝑥𝐵) → (𝐹𝐴) ⊆ 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257  wss 2945   class class class wbr 3792  cio 4893  cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-iota 4895  df-fv 4938
This theorem is referenced by:  fvssunirng  5218  relfvssunirn  5219  sefvex  5224  fvmptss2  5275  tfrexlem  5979
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