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Theorem fimadmfo 6766
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 6678 . 2 (𝐹:𝐴⟢𝐡 β†’ dom 𝐹 = 𝐴)
2 ffn 6669 . . . . 5 (𝐹:𝐴⟢𝐡 β†’ 𝐹 Fn 𝐴)
32adantr 482 . . . 4 ((𝐹:𝐴⟢𝐡 ∧ dom 𝐹 = 𝐴) β†’ 𝐹 Fn 𝐴)
4 dffn4 6763 . . . 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 6010 . . . . . . 7 (𝐴 = dom 𝐹 β†’ (𝐹 β€œ 𝐴) = (𝐹 β€œ dom 𝐹))
7 imadmrn 6024 . . . . . . 7 (𝐹 β€œ dom 𝐹) = ran 𝐹
86, 7eqtrdi 2793 . . . . . 6 (𝐴 = dom 𝐹 β†’ (𝐹 β€œ 𝐴) = ran 𝐹)
98eqcoms 2745 . . . . 5 (dom 𝐹 = 𝐴 β†’ (𝐹 β€œ 𝐴) = ran 𝐹)
109adantl 483 . . . 4 ((𝐹:𝐴⟢𝐡 ∧ dom 𝐹 = 𝐴) β†’ (𝐹 β€œ 𝐴) = ran 𝐹)
11 foeq3 6755 . . . 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 686 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 397   = wceq 1542  dom cdm 5634  ran crn 5635   β€œ cima 5637   Fn wfn 6492  βŸΆwf 6493  β€“ontoβ†’wfo 6495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fn 6500  df-f 6501  df-fo 6503
This theorem is referenced by:  wrdsymb  14431  fundcmpsurinjimaid  45610
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