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Theorem fvss 5132
 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 4853 . 2 (𝐹A) = (℩xA𝐹x)
2 iotass 4827 . 2 (x(A𝐹xxB) → (℩xA𝐹x) ⊆ B)
31, 2syl5eqss 2983 1 (x(A𝐹xxB) → (𝐹A) ⊆ B)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1240   ⊆ wss 2911   class class class wbr 3755  ℩cio 4808  ‘cfv 4845 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810  df-fv 4853 This theorem is referenced by:  fvssunirng  5133  relfvssunirn  5134  sefvex  5139  fvmptss2  5190  tfrexlem  5889
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