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Theorem fimadmfo 6572
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 6495 . 2 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2 ffn 6487 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
32adantr 484 . . . 4 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 Fn 𝐴)
4 dffn4 6569 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
53, 4sylib 221 . . 3 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴onto→ran 𝐹)
6 imaeq2 5898 . . . . . . 7 (𝐴 = dom 𝐹 → (𝐹𝐴) = (𝐹 “ dom 𝐹))
7 imadmrn 5912 . . . . . . 7 (𝐹 “ dom 𝐹) = ran 𝐹
86, 7syl6eq 2872 . . . . . 6 (𝐴 = dom 𝐹 → (𝐹𝐴) = ran 𝐹)
98eqcoms 2829 . . . . 5 (dom 𝐹 = 𝐴 → (𝐹𝐴) = ran 𝐹)
109adantl 485 . . . 4 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹𝐴) = ran 𝐹)
11 foeq3 6561 . . . 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 260 . 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 209  wa 399   = wceq 1538  dom cdm 5528  ran crn 5529  cima 5531   Fn wfn 6323  wf 6324  ontowfo 6326
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 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
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-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-xp 5534  df-cnv 5536  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-fn 6331  df-f 6332  df-fo 6334
This theorem is referenced by:  wrdsymb  13873  fundcmpsurinjimaid  43751
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