Theorem fvss 5500

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 5196	. 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
2		iotass 5170	. 2 ⊢ (∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ 𝐵) → (℩𝑥𝐴𝐹𝑥) ⊆ 𝐵)
3	1, 2	eqsstrid 3188	1 ⊢ (∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ 𝐵) → (𝐹‘𝐴) ⊆ 𝐵)

Syntax hints:   → wi 4  ∀wal 1341   ⊆ wss 3116   class class class wbr 3982  ℩cio 5151  ‘cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-iota 5153  df-fv 5196

This theorem is referenced by:  fvssunirng  5501  relfvssunirn  5502  sefvex  5507  fvmptss2  5561  tfrexlem  6302