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Theorem fimadmfoALT 6828
Description: Alternate proof of fimadmfo 6826, based on fores 6827. 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:𝐴𝐵𝐹:𝐴onto→(𝐹𝐴))

Proof of Theorem fimadmfoALT
StepHypRef Expression
1 fdm 6739 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2 frel 6735 . . . . 5 (𝐹:𝐴𝐵 → Rel 𝐹)
3 resdm 6037 . . . . . 6 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
43eqcomd 2732 . . . . 5 (Rel 𝐹𝐹 = (𝐹 ↾ dom 𝐹))
52, 4syl 17 . . . 4 (𝐹:𝐴𝐵𝐹 = (𝐹 ↾ dom 𝐹))
6 reseq2 5986 . . . 4 (dom 𝐹 = 𝐴 → (𝐹 ↾ dom 𝐹) = (𝐹𝐴))
75, 6sylan9eq 2786 . . 3 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 = (𝐹𝐴))
81, 7mpdan 685 . 2 (𝐹:𝐴𝐵𝐹 = (𝐹𝐴))
9 ffun 6733 . . . . . 6 (𝐹:𝐴𝐵 → Fun 𝐹)
10 eqimss2 4039 . . . . . . 7 (dom 𝐹 = 𝐴𝐴 ⊆ dom 𝐹)
111, 10syl 17 . . . . . 6 (𝐹:𝐴𝐵𝐴 ⊆ dom 𝐹)
129, 11jca 510 . . . . 5 (𝐹:𝐴𝐵 → (Fun 𝐹𝐴 ⊆ dom 𝐹))
1312adantr 479 . . . 4 ((𝐹:𝐴𝐵𝐹 = (𝐹𝐴)) → (Fun 𝐹𝐴 ⊆ dom 𝐹))
14 fores 6827 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
1513, 14syl 17 . . 3 ((𝐹:𝐴𝐵𝐹 = (𝐹𝐴)) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
16 foeq1 6813 . . . 4 (𝐹 = (𝐹𝐴) → (𝐹:𝐴onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴onto→(𝐹𝐴)))
1716adantl 480 . . 3 ((𝐹:𝐴𝐵𝐹 = (𝐹𝐴)) → (𝐹:𝐴onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴onto→(𝐹𝐴)))
1815, 17mpbird 256 . 2 ((𝐹:𝐴𝐵𝐹 = (𝐹𝐴)) → 𝐹:𝐴onto→(𝐹𝐴))
198, 18mpdan 685 1 (𝐹:𝐴𝐵𝐹:𝐴onto→(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wss 3947  dom cdm 5684  cres 5686  cima 5687  Rel wrel 5689  Fun wfun 6550  wf 6552  ontowfo 6554
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 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5156  df-opab 5218  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6558  df-fn 6559  df-f 6560  df-fo 6562
This theorem is referenced by: (None)
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