Theorem fvelimabd 6904

Description: Deduction form of fvelimab 6903. (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 6903	. 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 1541   ∈ wcel 2113  ∃wrex 3057   ⊆ wss 3898   “ cima 5624   Fn wfn 6484  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  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 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-fv 6497

This theorem is referenced by:  unima  6906  resf1extb  7873  ghmqusnsglem1  19200  ghmquskerlem1  19203  lmhmima  20990  mdegldg  26018  ig1peu  26127  2ndimaxp  32650  fnpreimac  32675  fsuppcurry1  32731  fsuppcurry2  32732  swrdrn3  32965  esplyfv1  33655  esplyfv  33656  esplyfval3  33658  fnrelpredd  35174  bj-gabima  37057  extoimad  44321  upgrimpths  48071