Theorem fvss 5602

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(𝐴𝐹𝑥 → 𝑥 ⊆ 𝐵) → (𝐹‘𝐴) ⊆ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem fvss

Step	Hyp	Ref	Expression
1		df-fv 5287	. 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
2		iotass 5257	. 2 ⊢ (∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ 𝐵) → (℩𝑥𝐴𝐹𝑥) ⊆ 𝐵)
3	1, 2	eqsstrid 3243	1 ⊢ (∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ 𝐵) → (𝐹‘𝐴) ⊆ 𝐵)

Syntax hints:   → wi 4  ∀wal 1371   ⊆ wss 3170   class class class wbr 4050  ℩cio 5238  ‘cfv 5279

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-uni 3856  df-iota 5240  df-fv 5287

This theorem is referenced by:  fvssunirng  5603  relfvssunirn  5604  sefvex  5609  fvmptss2  5666  tfrexlem  6432