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 Description: Alternate proof of fimadmfo 6590, based on fores 6591. 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

StepHypRef Expression
1 fdm 6511 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2 frel 6508 . . . . 5 (𝐹:𝐴𝐵 → Rel 𝐹)
3 resdm 5884 . . . . . 6 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
43eqcomd 2830 . . . . 5 (Rel 𝐹𝐹 = (𝐹 ↾ dom 𝐹))
52, 4syl 17 . . . 4 (𝐹:𝐴𝐵𝐹 = (𝐹 ↾ dom 𝐹))
6 reseq2 5835 . . . 4 (dom 𝐹 = 𝐴 → (𝐹 ↾ dom 𝐹) = (𝐹𝐴))
75, 6sylan9eq 2879 . . 3 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 = (𝐹𝐴))
81, 7mpdan 686 . 2 (𝐹:𝐴𝐵𝐹 = (𝐹𝐴))
9 ffun 6506 . . . . . 6 (𝐹:𝐴𝐵 → Fun 𝐹)
10 eqimss2 4010 . . . . . . 7 (dom 𝐹 = 𝐴𝐴 ⊆ dom 𝐹)
111, 10syl 17 . . . . . 6 (𝐹:𝐴𝐵𝐴 ⊆ dom 𝐹)
129, 11jca 515 . . . . 5 (𝐹:𝐴𝐵 → (Fun 𝐹𝐴 ⊆ dom 𝐹))
1312adantr 484 . . . 4 ((𝐹:𝐴𝐵𝐹 = (𝐹𝐴)) → (Fun 𝐹𝐴 ⊆ dom 𝐹))
14 fores 6591 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
1513, 14syl 17 . . 3 ((𝐹:𝐴𝐵𝐹 = (𝐹𝐴)) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
16 foeq1 6577 . . . 4 (𝐹 = (𝐹𝐴) → (𝐹:𝐴onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴onto→(𝐹𝐴)))
1716adantl 485 . . 3 ((𝐹:𝐴𝐵𝐹 = (𝐹𝐴)) → (𝐹:𝐴onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴onto→(𝐹𝐴)))
1815, 17mpbird 260 . 2 ((𝐹:𝐴𝐵𝐹 = (𝐹𝐴)) → 𝐹:𝐴onto→(𝐹𝐴))
198, 18mpdan 686 1 (𝐹:𝐴𝐵𝐹:𝐴onto→(𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ⊆ wss 3919  dom cdm 5542   ↾ cres 5544   “ cima 5545  Rel wrel 5547  Fun wfun 6337  ⟶wf 6339  –onto→wfo 6341 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-fun 6345  df-fn 6346  df-f 6347  df-fo 6349 This theorem is referenced by: (None)
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