Theorem fvelimabd 6907

Description: Deduction form of fvelimab 6906. (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 6906	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))
4	1, 2, 3	syl2anc 585	1 ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3890   “ cima 5627   Fn wfn 6487  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by:  unima  6909  resf1extb  7878  ghmqusnsglem1  19246  ghmquskerlem1  19249  lmhmima  21034  mdegldg  26041  ig1peu  26150  2ndimaxp  32734  fnpreimac  32758  fsuppcurry1  32812  fsuppcurry2  32813  swrdrn3  33030  esplyfv1  33728  esplyfv  33729  esplyfval3  33731  fnrelpredd  35250  bj-gabima  37263  extoimad  44609  upgrimpths  48397