Theorem fimadmfo 5577

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 5495	. 2 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
2		ffn 5489	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3	2	adantr 276	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 Fn 𝐴)
4		dffn4 5574	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)
5	3, 4	sylib 122	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴–onto→ran 𝐹)
6		imaeq2 5078	. . . . . . 7 ⊢ (𝐴 = dom 𝐹 → (𝐹 “ 𝐴) = (𝐹 “ dom 𝐹))
7		imadmrn 5092	. . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
8	6, 7	eqtrdi 2280	. . . . . 6 ⊢ (𝐴 = dom 𝐹 → (𝐹 “ 𝐴) = ran 𝐹)
9	8	eqcoms 2234	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
10	9	adantl 277	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹 “ 𝐴) = ran 𝐹)
11		foeq3 5566	. . . 4 ⊢ ((𝐹 “ 𝐴) = ran 𝐹 → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ 𝐹:𝐴–onto→ran 𝐹))
12	10, 11	syl 14	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ 𝐹:𝐴–onto→ran 𝐹))
13	5, 12	mpbird 167	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴–onto→(𝐹 “ 𝐴))
14	1, 13	mpdan 421	1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  dom cdm 4731  ran crn 4732   “ cima 4734   Fn wfn 5328  ⟶wf 5329  –onto→wfo 5331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-fn 5336  df-f 5337  df-fo 5339

This theorem is referenced by:  wrdsymb  11188