MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fimadmfo Structured version   Visualization version   GIF version

Theorem fimadmfo 6366
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 6290 . 2 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2 ffn 6282 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
32adantr 474 . . . 4 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 Fn 𝐴)
4 dffn4 6363 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
53, 4sylib 210 . . 3 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴onto→ran 𝐹)
6 imaeq2 5707 . . . . . . 7 (𝐴 = dom 𝐹 → (𝐹𝐴) = (𝐹 “ dom 𝐹))
7 imadmrn 5721 . . . . . . 7 (𝐹 “ dom 𝐹) = ran 𝐹
86, 7syl6eq 2877 . . . . . 6 (𝐴 = dom 𝐹 → (𝐹𝐴) = ran 𝐹)
98eqcoms 2833 . . . . 5 (dom 𝐹 = 𝐴 → (𝐹𝐴) = ran 𝐹)
109adantl 475 . . . 4 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → (𝐹𝐴) = ran 𝐹)
11 foeq3 6355 . . . 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 249 . 2 ((𝐹:𝐴𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹:𝐴onto→(𝐹𝐴))
141, 13mpdan 678 1 (𝐹:𝐴𝐵𝐹:𝐴onto→(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  dom cdm 5346  ran crn 5347  cima 5349   Fn wfn 6122  wf 6123  ontowfo 6125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-cnv 5354  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-fn 6130  df-f 6131  df-fo 6133
This theorem is referenced by:  wrdsymb  13609
  Copyright terms: Public domain W3C validator