Theorem fvelimabd 6900

Description: Deduction form of fvelimab 6899. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypotheses

Ref	Expression
fvelimabd.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
fvelimabd.2	⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Assertion

Ref	Expression
fvelimabd	⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))

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

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem fvelimabd

Step	Hyp	Ref	Expression
1		fvelimabd.1	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		fvelimabd.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
3		fvelimab 6899	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3905   “ cima 5626   Fn wfn 6481  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494

This theorem is referenced by:  unima  6902  resf1extb  7874  ghmqusnsglem1  19177  ghmquskerlem1  19180  lmhmima  20969  mdegldg  25987  ig1peu  26096  2ndimaxp  32603  fnpreimac  32628  fsuppcurry1  32681  fsuppcurry2  32682  swrdrn3  32910  fnrelpredd  35058  bj-gabima  36916  extoimad  44140  upgrimpths  47897