Theorem fvss 5572

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 5266	. 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
2		iotass 5236	. 2 ⊢ (∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ 𝐵) → (℩𝑥𝐴𝐹𝑥) ⊆ 𝐵)
3	1, 2	eqsstrid 3229	1 ⊢ (∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ 𝐵) → (𝐹‘𝐴) ⊆ 𝐵)

Syntax hints:   → wi 4  ∀wal 1362   ⊆ wss 3157   class class class wbr 4033  ℩cio 5217  ‘cfv 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-iota 5219  df-fv 5266

This theorem is referenced by:  fvssunirng  5573  relfvssunirn  5574  sefvex  5579  fvmptss2  5636  tfrexlem  6392