Theorem fimadmfoALT 6801

Description: Alternate proof of fimadmfo 6799, based on fores 6800. A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
fimadmfoALT	⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴))

Proof of Theorem fimadmfoALT

Step	Hyp	Ref	Expression
1		fdm 6713	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
2		frel 6709	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹)
3		resdm 6023	. . . . . 6 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
4	3	eqcomd 2775	. . . . 5 ⊢ (Rel 𝐹 → 𝐹 = (𝐹 ↾ dom 𝐹))
5	2, 4	syl 18	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝐹 ↾ dom 𝐹))
6		reseq2 5971	. . . 4 ⊢ (dom 𝐹 = 𝐴 → (𝐹 ↾ dom 𝐹) = (𝐹 ↾ 𝐴))
7	5, 6	sylan9eq 2824	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 = (𝐹 ↾ 𝐴))
8	1, 7	mpdan 699	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝐹 ↾ 𝐴))
9		ffun 6706	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
10		eqimss2 4004	. . . . . . 7 ⊢ (dom 𝐹 = 𝐴 → 𝐴 ⊆ dom 𝐹)
11	1, 10	syl 18	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹)
12	9, 11	jca 520	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹))
13	12	adantr 485	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹))
14		fores 6800	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))
15	13, 14	syl 18	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))
16		foeq1 6786	. . . 4 ⊢ (𝐹 = (𝐹 ↾ 𝐴) → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)))
17	16	adantl 486	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)))
18	15, 17	mpbird 260	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → 𝐹:𝐴–onto→(𝐹 “ 𝐴))
19	8, 18	mpdan 699	1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ⊆ wss 3913  dom cdm 5659   ↾ cres 5661   “ cima 5662  Rel wrel 5664  Fun wfun 6528  ⟶wf 6530  –onto→wfo 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6536  df-fn 6537  df-f 6538  df-fo 6540

This theorem is referenced by: (None)