Theorem fndmexb 7850

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 7165	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ V) → 𝐹 ∈ V)
2	1	ex 414	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐴 ∈ V → 𝐹 ∈ V))
3		simpr 486	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
4		simpl 484	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ V) → 𝐹 Fn 𝐴)
5	3, 4	fndmexd 7848	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ V) → 𝐴 ∈ V)
6	5	ex 414	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∈ V → 𝐴 ∈ V))
7	2, 6	impbid 214	1 ⊢ (𝐹 Fn 𝐴 → (𝐴 ∈ V ↔ 𝐹 ∈ V))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∈ wcel 2121  Vcvv 3433   Fn wfn 6484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497

This theorem is referenced by:  fdmexb  7851