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Theorem fimadmfo 6808
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 6720 . 2 (𝐹:𝐴⟢𝐡 β†’ dom 𝐹 = 𝐴)
2 ffn 6711 . . . . 5 (𝐹:𝐴⟢𝐡 β†’ 𝐹 Fn 𝐴)
32adantr 480 . . . 4 ((𝐹:𝐴⟢𝐡 ∧ dom 𝐹 = 𝐴) β†’ 𝐹 Fn 𝐴)
4 dffn4 6805 . . . 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 6049 . . . . . . 7 (𝐴 = dom 𝐹 β†’ (𝐹 β€œ 𝐴) = (𝐹 β€œ dom 𝐹))
7 imadmrn 6063 . . . . . . 7 (𝐹 β€œ dom 𝐹) = ran 𝐹
86, 7eqtrdi 2782 . . . . . 6 (𝐴 = dom 𝐹 β†’ (𝐹 β€œ 𝐴) = ran 𝐹)
98eqcoms 2734 . . . . 5 (dom 𝐹 = 𝐴 β†’ (𝐹 β€œ 𝐴) = ran 𝐹)
109adantl 481 . . . 4 ((𝐹:𝐴⟢𝐡 ∧ dom 𝐹 = 𝐴) β†’ (𝐹 β€œ 𝐴) = ran 𝐹)
11 foeq3 6797 . . . 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 257 . 2 ((𝐹:𝐴⟢𝐡 ∧ dom 𝐹 = 𝐴) β†’ 𝐹:𝐴–ontoβ†’(𝐹 β€œ 𝐴))
141, 13mpdan 684 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 395   = wceq 1533  dom cdm 5669  ran crn 5670   β€œ cima 5672   Fn wfn 6532  βŸΆwf 6533  β€“ontoβ†’wfo 6535
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-fn 6540  df-f 6541  df-fo 6543
This theorem is referenced by:  wrdsymb  14498  imasmhm  32972  imasghm  32973  imasrhm  32974  imaslmhm  32975  r1pquslmic  33186  fundcmpsurinjimaid  46651
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