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Theorem fimadmfo 5477
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 5401 . 2 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2 ffn 5395 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
32adantr 276 . . . 4 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 Fn 𝐴)
4 dffn4 5474 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
53, 4sylib 122 . . 3 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴onto→ran 𝐹)
6 imaeq2 4995 . . . . . . 7 (𝐴 = dom 𝐹 → (𝐹𝐴) = (𝐹 “ dom 𝐹))
7 imadmrn 5009 . . . . . . 7 (𝐹 “ dom 𝐹) = ran 𝐹
86, 7eqtrdi 2242 . . . . . 6 (𝐴 = dom 𝐹 → (𝐹𝐴) = ran 𝐹)
98eqcoms 2196 . . . . 5 (dom 𝐹 = 𝐴 → (𝐹𝐴) = ran 𝐹)
109adantl 277 . . . 4 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹𝐴) = ran 𝐹)
11 foeq3 5466 . . . 4 ((𝐹𝐴) = ran 𝐹 → (𝐹:𝐴onto→(𝐹𝐴) ↔ 𝐹:𝐴onto→ran 𝐹))
1210, 11syl 14 . . 3 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹:𝐴onto→(𝐹𝐴) ↔ 𝐹:𝐴onto→ran 𝐹))
135, 12mpbird 167 . 2 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴onto→(𝐹𝐴))
141, 13mpdan 421 1 (𝐹:𝐴𝐵𝐹:𝐴onto→(𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  dom cdm 4655  ran crn 4656  cima 4658   Fn wfn 5241  wf 5242  ontowfo 5244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4661  df-cnv 4663  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fn 5249  df-f 5250  df-fo 5252
This theorem is referenced by:  wrdsymb  10931
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