Theorem fimadmfo 6805

Description: A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.)

Assertion

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

Proof of Theorem fimadmfo

Step	Hyp	Ref	Expression
1		fdm 6719	. 2 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
2		ffn 6709	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3	2	adantr 486	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 Fn 𝐴)
4		dffn4 6802	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)
5	3, 4	sylib 221	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴–onto→ran 𝐹)
6		imaeq2 6060	. . . . . . 7 ⊢ (𝐴 = dom 𝐹 → (𝐹 “ 𝐴) = (𝐹 “ dom 𝐹))
7		imadmrn 6074	. . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
8	6, 7	eqtrdi 2816	. . . . . 6 ⊢ (𝐴 = dom 𝐹 → (𝐹 “ 𝐴) = ran 𝐹)
9	8	eqcoms 2773	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
10	9	adantl 487	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹 “ 𝐴) = ran 𝐹)
11		foeq3 6794	. . . 4 ⊢ ((𝐹 “ 𝐴) = ran 𝐹 → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ 𝐹:𝐴–onto→ran 𝐹))
12	10, 11	syl 18	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ 𝐹:𝐴–onto→ran 𝐹))
13	5, 12	mpbird 260	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴–onto→(𝐹 “ 𝐴))
14	1, 13	mpdan 700	1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  dom cdm 5663  ran crn 5664   “ cima 5666   Fn wfn 6535  ⟶wf 6536  –onto→wfo 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-fn 6543  df-f 6544  df-fo 6546

This theorem is used by:  wrdsymb  14593  imasmhm  33714  imasghm  33715  imasrhm  33716  imaslmhm  33717  r1pquslmic  33940  fundcmpsurinjimaid  48193