Theorem feldmfvelcdm 7042

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 7040	. . 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 3038	. . . 4 ⊢ (∅ ∉ 𝐵 ↔ ¬ ∅ ∈ 𝐵)
5		nelelne 3032	. . . 4 ⊢ (¬ ∅ ∈ 𝐵 → ((𝐹‘𝑋) ∈ 𝐵 → (𝐹‘𝑋) ≠ ∅))
6	4, 5	sylbi 217	. . 3 ⊢ (∅ ∉ 𝐵 → ((𝐹‘𝑋) ∈ 𝐵 → (𝐹‘𝑋) ≠ ∅))
7		fdm 6681	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
8		fvfundmfvn0 6884	. . . 4 ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝑋 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑋})))
9		simprl 771	. . . . . 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 2839	. . . . 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 1542   ∈ wcel 2114   ≠ wne 2933   ∉ wnel 3037  ∅c0 4287  {csn 4582  dom cdm 5634   ↾ cres 5636  Fun wfun 6496  ⟶wf 6498  ‘cfv 6502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-fv 6510

This theorem is referenced by: (None)