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Theorem fimadmfoALT 6816
Description: Alternate proof of fimadmfo 6814, based on fores 6815. 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 6725 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2 frel 6721 . . . . 5 (𝐹:𝐴𝐵 → Rel 𝐹)
3 resdm 6024 . . . . . 6 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
43eqcomd 2733 . . . . 5 (Rel 𝐹𝐹 = (𝐹 ↾ dom 𝐹))
52, 4syl 17 . . . 4 (𝐹:𝐴𝐵𝐹 = (𝐹 ↾ dom 𝐹))
6 reseq2 5974 . . . 4 (dom 𝐹 = 𝐴 → (𝐹 ↾ dom 𝐹) = (𝐹𝐴))
75, 6sylan9eq 2787 . . 3 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 = (𝐹𝐴))
81, 7mpdan 686 . 2 (𝐹:𝐴𝐵𝐹 = (𝐹𝐴))
9 ffun 6719 . . . . . 6 (𝐹:𝐴𝐵 → Fun 𝐹)
10 eqimss2 4037 . . . . . . 7 (dom 𝐹 = 𝐴𝐴 ⊆ dom 𝐹)
111, 10syl 17 . . . . . 6 (𝐹:𝐴𝐵𝐴 ⊆ dom 𝐹)
129, 11jca 511 . . . . 5 (𝐹:𝐴𝐵 → (Fun 𝐹𝐴 ⊆ dom 𝐹))
1312adantr 480 . . . 4 ((𝐹:𝐴𝐵𝐹 = (𝐹𝐴)) → (Fun 𝐹𝐴 ⊆ dom 𝐹))
14 fores 6815 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
1513, 14syl 17 . . 3 ((𝐹:𝐴𝐵𝐹 = (𝐹𝐴)) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
16 foeq1 6801 . . . 4 (𝐹 = (𝐹𝐴) → (𝐹:𝐴onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴onto→(𝐹𝐴)))
1716adantl 481 . . 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 395   = wceq 1534  wss 3944  dom cdm 5672  cres 5674  cima 5675  Rel wrel 5677  Fun wfun 6536  wf 6538  ontowfo 6540
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 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
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 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548
This theorem is referenced by: (None)
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