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Theorem fimadmfo 6820
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 6731 . 2 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2 ffn 6722 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
32adantr 480 . . . 4 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 Fn 𝐴)
4 dffn4 6817 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
53, 4sylib 217 . . 3 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴onto→ran 𝐹)
6 imaeq2 6059 . . . . . . 7 (𝐴 = dom 𝐹 → (𝐹𝐴) = (𝐹 “ dom 𝐹))
7 imadmrn 6073 . . . . . . 7 (𝐹 “ dom 𝐹) = ran 𝐹
86, 7eqtrdi 2784 . . . . . 6 (𝐴 = dom 𝐹 → (𝐹𝐴) = ran 𝐹)
98eqcoms 2736 . . . . 5 (dom 𝐹 = 𝐴 → (𝐹𝐴) = ran 𝐹)
109adantl 481 . . . 4 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹𝐴) = ran 𝐹)
11 foeq3 6809 . . . 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 686 1 (𝐹:𝐴𝐵𝐹:𝐴onto→(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  dom cdm 5678  ran crn 5679  cima 5681   Fn wfn 6543  wf 6544  ontowfo 6546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-fn 6551  df-f 6552  df-fo 6554
This theorem is referenced by:  wrdsymb  14525  imasmhm  33079  imasghm  33080  imasrhm  33081  imaslmhm  33082  r1pquslmic  33281  fundcmpsurinjimaid  46751
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