MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fimadmfo Structured version   Visualization version   GIF version

Theorem fimadmfo 6829
Description: A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.)
Assertion
Ref Expression
fimadmfo (𝐹:𝐴𝐵𝐹:𝐴onto→(𝐹𝐴))

Proof of Theorem fimadmfo
StepHypRef Expression
1 fdm 6745 . 2 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2 ffn 6736 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
32adantr 480 . . . 4 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 Fn 𝐴)
4 dffn4 6826 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
53, 4sylib 218 . . 3 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴onto→ran 𝐹)
6 imaeq2 6074 . . . . . . 7 (𝐴 = dom 𝐹 → (𝐹𝐴) = (𝐹 “ dom 𝐹))
7 imadmrn 6088 . . . . . . 7 (𝐹 “ dom 𝐹) = ran 𝐹
86, 7eqtrdi 2793 . . . . . 6 (𝐴 = dom 𝐹 → (𝐹𝐴) = ran 𝐹)
98eqcoms 2745 . . . . 5 (dom 𝐹 = 𝐴 → (𝐹𝐴) = ran 𝐹)
109adantl 481 . . . 4 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹𝐴) = ran 𝐹)
11 foeq3 6818 . . . 4 ((𝐹𝐴) = ran 𝐹 → (𝐹:𝐴onto→(𝐹𝐴) ↔ 𝐹:𝐴onto→ran 𝐹))
1210, 11syl 17 . . 3 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹:𝐴onto→(𝐹𝐴) ↔ 𝐹:𝐴onto→ran 𝐹))
135, 12mpbird 257 . 2 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴onto→(𝐹𝐴))
141, 13mpdan 687 1 (𝐹:𝐴𝐵𝐹:𝐴onto→(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  dom cdm 5685  ran crn 5686  cima 5688   Fn wfn 6556  wf 6557  ontowfo 6559
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fn 6564  df-f 6565  df-fo 6567
This theorem is referenced by:  wrdsymb  14580  imasmhm  33382  imasghm  33383  imasrhm  33384  imaslmhm  33385  r1pquslmic  33631  fundcmpsurinjimaid  47398
  Copyright terms: Public domain W3C validator