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Theorem fimadmfoALT 6768
Description: Alternate proof of fimadmfo 6766, based on fores 6767. 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 6678 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2 frel 6674 . . . . 5 (𝐹:𝐴𝐵 → Rel 𝐹)
3 resdm 5983 . . . . . 6 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
43eqcomd 2743 . . . . 5 (Rel 𝐹𝐹 = (𝐹 ↾ dom 𝐹))
52, 4syl 17 . . . 4 (𝐹:𝐴𝐵𝐹 = (𝐹 ↾ dom 𝐹))
6 reseq2 5933 . . . 4 (dom 𝐹 = 𝐴 → (𝐹 ↾ dom 𝐹) = (𝐹𝐴))
75, 6sylan9eq 2797 . . 3 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 = (𝐹𝐴))
81, 7mpdan 686 . 2 (𝐹:𝐴𝐵𝐹 = (𝐹𝐴))
9 ffun 6672 . . . . . 6 (𝐹:𝐴𝐵 → Fun 𝐹)
10 eqimss2 4002 . . . . . . 7 (dom 𝐹 = 𝐴𝐴 ⊆ dom 𝐹)
111, 10syl 17 . . . . . 6 (𝐹:𝐴𝐵𝐴 ⊆ dom 𝐹)
129, 11jca 513 . . . . 5 (𝐹:𝐴𝐵 → (Fun 𝐹𝐴 ⊆ dom 𝐹))
1312adantr 482 . . . 4 ((𝐹:𝐴𝐵𝐹 = (𝐹𝐴)) → (Fun 𝐹𝐴 ⊆ dom 𝐹))
14 fores 6767 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
1513, 14syl 17 . . 3 ((𝐹:𝐴𝐵𝐹 = (𝐹𝐴)) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
16 foeq1 6753 . . . 4 (𝐹 = (𝐹𝐴) → (𝐹:𝐴onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴onto→(𝐹𝐴)))
1716adantl 483 . . 3 ((𝐹:𝐴𝐵𝐹 = (𝐹𝐴)) → (𝐹:𝐴onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴onto→(𝐹𝐴)))
1815, 17mpbird 257 . 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 205  wa 397   = wceq 1542  wss 3911  dom cdm 5634  cres 5636  cima 5637  Rel wrel 5639  Fun wfun 6491  wf 6493  ontowfo 6495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503
This theorem is referenced by: (None)
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