Theorem fvn0fvelrnOLD 7097

Description: Obsolete version of fvn0fvelrn 6851 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 6863	. 2 ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝑋 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑋})))
2		eldmressnsn 5975	. . . 4 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ dom (𝐹 ↾ {𝑋}))
3		fvelrn 7010	. . . . . . 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 7096	. . . . 5 ⊢ (𝑋 ∈ dom 𝐹 → ((𝐹 ↾ {𝑋})‘𝑋) = (𝐹‘𝑋))
11	10	eleq1d 2813	. . . 4 ⊢ (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ↔ (𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋})))
12		fvrnressn 7095	. . . 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 2109   ≠ wne 2925  ∅c0 4284  {csn 4577  dom cdm 5619  ran crn 5620   ↾ cres 5621  Fun wfun 6476  ‘cfv 6482

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 5235  ax-nul 5245  ax-pr 5371

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-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-fv 6490

This theorem is referenced by: (None)