Theorem fndmexb 7918

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 7233	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ V) → 𝐹 ∈ V)
2	1	ex 411	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐴 ∈ V → 𝐹 ∈ V))
3		simpr 483	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
4		simpl 481	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ V) → 𝐹 Fn 𝐴)
5	3, 4	fndmexd 7916	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ V) → 𝐴 ∈ V)
6	5	ex 411	. 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 394   ∈ wcel 2098  Vcvv 3471   Fn wfn 6546

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 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559

This theorem is referenced by:  fdmexb  7919