Theorem fvn0fvelrnOLD 7183

Description: Obsolete version of fvn0fvelrn 6938 as of 13-Jan-2025. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
fvn0fvelrnOLD	⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹)

Proof of Theorem fvn0fvelrnOLD

Step	Hyp	Ref	Expression
1		fvfundmfvn0 6950	. 2 ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝑋 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑋})))
2		eldmressnsn 6044	. . . 4 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ dom (𝐹 ↾ {𝑋}))
3		fvelrn 7096	. . . . . . 7 ⊢ ((Fun (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom (𝐹 ↾ {𝑋})) → ((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}))
4		pm3.2 469	. . . . . . 7 ⊢ (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹)))
5	3, 4	syl 17	. . . . . 6 ⊢ ((Fun (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom (𝐹 ↾ {𝑋})) → (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹)))
6	5	ex 412	. . . . 5 ⊢ (Fun (𝐹 ↾ {𝑋}) → (𝑋 ∈ dom (𝐹 ↾ {𝑋}) → (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹))))
7	6	com13 88	. . . 4 ⊢ (𝑋 ∈ dom 𝐹 → (𝑋 ∈ dom (𝐹 ↾ {𝑋}) → (Fun (𝐹 ↾ {𝑋}) → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹))))
8	2, 7	mpd 15	. . 3 ⊢ (𝑋 ∈ dom 𝐹 → (Fun (𝐹 ↾ {𝑋}) → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹)))
9	8	imp 406	. 2 ⊢ ((𝑋 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑋})) → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹))
10		fvressn 7182	. . . . 5 ⊢ (𝑋 ∈ dom 𝐹 → ((𝐹 ↾ {𝑋})‘𝑋) = (𝐹‘𝑋))
11	10	eleq1d 2824	. . . 4 ⊢ (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ↔ (𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋})))
12		fvrnressn 7181	. . . 4 ⊢ (𝑋 ∈ dom 𝐹 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹))
13	11, 12	sylbid 240	. . 3 ⊢ (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹))
14	13	impcom 407	. 2 ⊢ ((((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) ∈ ran 𝐹)
15	1, 9, 14	3syl 18	1 ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2106   ≠ wne 2938  ∅c0 4339  {csn 4631  dom cdm 5689  ran crn 5690   ↾ cres 5691  Fun wfun 6557  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571

This theorem is referenced by: (None)