Theorem feldmfvelcdm 7040

Description: A class is an element of the domain iff it's function value is an element of the codomain of a function. (Contributed by AV, 22-Apr-2025.)

Assertion

Ref	Expression
feldmfvelcdm	⊢ ((𝐹:𝐴⟶𝐵 ∧ ∅ ∉ 𝐵) → (𝑋 ∈ 𝐴 ↔ (𝐹‘𝑋) ∈ 𝐵))

Proof of Theorem feldmfvelcdm

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∅ ∉ 𝐵) → 𝐹:𝐴⟶𝐵)
2	1	ffvelcdmda 7038	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ ∅ ∉ 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ 𝐵)
3	2	ex 412	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∅ ∉ 𝐵) → (𝑋 ∈ 𝐴 → (𝐹‘𝑋) ∈ 𝐵))
4		df-nel 3030	. . . 4 ⊢ (∅ ∉ 𝐵 ↔ ¬ ∅ ∈ 𝐵)
5		nelelne 3024	. . . 4 ⊢ (¬ ∅ ∈ 𝐵 → ((𝐹‘𝑋) ∈ 𝐵 → (𝐹‘𝑋) ≠ ∅))
6	4, 5	sylbi 217	. . 3 ⊢ (∅ ∉ 𝐵 → ((𝐹‘𝑋) ∈ 𝐵 → (𝐹‘𝑋) ≠ ∅))
7		fdm 6679	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
8		fvfundmfvn0 6883	. . . 4 ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝑋 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑋})))
9		simprl 770	. . . . . 6 ⊢ ((dom 𝐹 = 𝐴 ∧ (𝑋 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑋}))) → 𝑋 ∈ dom 𝐹)
10		simpl 482	. . . . . 6 ⊢ ((dom 𝐹 = 𝐴 ∧ (𝑋 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑋}))) → dom 𝐹 = 𝐴)
11	9, 10	eleqtrd 2830	. . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ (𝑋 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑋}))) → 𝑋 ∈ 𝐴)
12	11	ex 412	. . . 4 ⊢ (dom 𝐹 = 𝐴 → ((𝑋 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑋})) → 𝑋 ∈ 𝐴))
13	7, 8, 12	syl2im 40	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹‘𝑋) ≠ ∅ → 𝑋 ∈ 𝐴))
14	6, 13	sylan9r 508	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∅ ∉ 𝐵) → ((𝐹‘𝑋) ∈ 𝐵 → 𝑋 ∈ 𝐴))
15	3, 14	impbid 212	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∅ ∉ 𝐵) → (𝑋 ∈ 𝐴 ↔ (𝐹‘𝑋) ∈ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ∉ wnel 3029  ∅c0 4292  {csn 4585  dom cdm 5631   ↾ cres 5633  Fun wfun 6493  ⟶wf 6495  ‘cfv 6499

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-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507

This theorem is referenced by: (None)