Theorem fndmexb 7895

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 7213	. . 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 7893	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ V) → 𝐴 ∈ V)
6	5	ex 412	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∈ V → 𝐴 ∈ V))
7	2, 6	impbid 211	1 ⊢ (𝐹 Fn 𝐴 → (𝐴 ∈ V ↔ 𝐹 ∈ V))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2098  Vcvv 3468   Fn wfn 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by:  fdmexb  7896