Theorem fvn0fvelrnOLD 7161

Description: Obsolete version of fvn0fvelrn 6923 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 6935	. 2 ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝑋 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑋})))
2		eldmressnsn 6025	. . . 4 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ dom (𝐹 ↾ {𝑋}))
3		fvelrn 7079	. . . . . . 7 ⊢ ((Fun (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom (𝐹 ↾ {𝑋})) → ((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}))
4		pm3.2 471	. . . . . . 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 414	. . . . 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 408	. 2 ⊢ ((𝑋 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑋})) → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹))
10		fvressn 7160	. . . . 5 ⊢ (𝑋 ∈ dom 𝐹 → ((𝐹 ↾ {𝑋})‘𝑋) = (𝐹‘𝑋))
11	10	eleq1d 2819	. . . 4 ⊢ (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ↔ (𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋})))
12		fvrnressn 7159	. . . 4 ⊢ (𝑋 ∈ dom 𝐹 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹))
13	11, 12	sylbid 239	. . 3 ⊢ (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹))
14	13	impcom 409	. 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 397   ∈ wcel 2107   ≠ wne 2941  ∅c0 4323  {csn 4629  dom cdm 5677  ran crn 5678   ↾ cres 5679  Fun wfun 6538  ‘cfv 6544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552

This theorem is referenced by: (None)