Theorem fndmexb 7929

Description: The domain of a function is a set iff the function is a set. (Contributed by AV, 8-Aug-2024.)

Assertion

Ref	Expression
fndmexb	⊢ (𝐹 Fn 𝐴 → (𝐴 ∈ V ↔ 𝐹 ∈ V))

Proof of Theorem fndmexb

Step	Hyp	Ref	Expression
1		fnex 7237	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ V) → 𝐹 ∈ V)
2	1	ex 412	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐴 ∈ V → 𝐹 ∈ V))
3		simpr 484	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
4		simpl 482	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ V) → 𝐹 Fn 𝐴)
5	3, 4	fndmexd 7927	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ V) → 𝐴 ∈ V)
6	5	ex 412	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∈ V → 𝐴 ∈ V))
7	2, 6	impbid 212	1 ⊢ (𝐹 Fn 𝐴 → (𝐴 ∈ V ↔ 𝐹 ∈ V))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2106  Vcvv 3478   Fn wfn 6558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571

This theorem is referenced by:  fdmexb  7930