Theorem fvelimabd 6937

Description: Deduction form of fvelimab 6936. (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 6936	. 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 3054   ⊆ wss 3917   “ cima 5644   Fn wfn 6509  ‘cfv 6514

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522

This theorem is referenced by:  unima  6939  resf1extb  7913  ghmqusnsglem1  19219  ghmquskerlem1  19222  lmhmima  20961  mdegldg  25978  ig1peu  26087  2ndimaxp  32577  fnpreimac  32602  fsuppcurry1  32655  fsuppcurry2  32656  swrdrn3  32884  fnrelpredd  35086  bj-gabima  36935  extoimad  44160  upgrimpths  47913