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Theorem fimadmfo 6825
Description: A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.)
Assertion
Ref Expression
fimadmfo (𝐹:𝐴⟢𝐡 β†’ 𝐹:𝐴–ontoβ†’(𝐹 β€œ 𝐴))

Proof of Theorem fimadmfo
StepHypRef Expression
1 fdm 6736 . 2 (𝐹:𝐴⟢𝐡 β†’ dom 𝐹 = 𝐴)
2 ffn 6727 . . . . 5 (𝐹:𝐴⟢𝐡 β†’ 𝐹 Fn 𝐴)
32adantr 479 . . . 4 ((𝐹:𝐴⟢𝐡 ∧ dom 𝐹 = 𝐴) β†’ 𝐹 Fn 𝐴)
4 dffn4 6822 . . . 4 (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–ontoβ†’ran 𝐹)
53, 4sylib 217 . . 3 ((𝐹:𝐴⟢𝐡 ∧ dom 𝐹 = 𝐴) β†’ 𝐹:𝐴–ontoβ†’ran 𝐹)
6 imaeq2 6064 . . . . . . 7 (𝐴 = dom 𝐹 β†’ (𝐹 β€œ 𝐴) = (𝐹 β€œ dom 𝐹))
7 imadmrn 6078 . . . . . . 7 (𝐹 β€œ dom 𝐹) = ran 𝐹
86, 7eqtrdi 2784 . . . . . 6 (𝐴 = dom 𝐹 β†’ (𝐹 β€œ 𝐴) = ran 𝐹)
98eqcoms 2736 . . . . 5 (dom 𝐹 = 𝐴 β†’ (𝐹 β€œ 𝐴) = ran 𝐹)
109adantl 480 . . . 4 ((𝐹:𝐴⟢𝐡 ∧ dom 𝐹 = 𝐴) β†’ (𝐹 β€œ 𝐴) = ran 𝐹)
11 foeq3 6814 . . . 4 ((𝐹 β€œ 𝐴) = ran 𝐹 β†’ (𝐹:𝐴–ontoβ†’(𝐹 β€œ 𝐴) ↔ 𝐹:𝐴–ontoβ†’ran 𝐹))
1210, 11syl 17 . . 3 ((𝐹:𝐴⟢𝐡 ∧ dom 𝐹 = 𝐴) β†’ (𝐹:𝐴–ontoβ†’(𝐹 β€œ 𝐴) ↔ 𝐹:𝐴–ontoβ†’ran 𝐹))
135, 12mpbird 256 . 2 ((𝐹:𝐴⟢𝐡 ∧ dom 𝐹 = 𝐴) β†’ 𝐹:𝐴–ontoβ†’(𝐹 β€œ 𝐴))
141, 13mpdan 685 1 (𝐹:𝐴⟢𝐡 β†’ 𝐹:𝐴–ontoβ†’(𝐹 β€œ 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  dom cdm 5682  ran crn 5683   β€œ cima 5685   Fn wfn 6548  βŸΆwf 6549  β€“ontoβ†’wfo 6551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-fn 6556  df-f 6557  df-fo 6559
This theorem is referenced by:  wrdsymb  14534  imasmhm  33098  imasghm  33099  imasrhm  33100  imaslmhm  33101  r1pquslmic  33322  fundcmpsurinjimaid  46798
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