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Theorem focnvimacdmdm 6769
Description: The preimage of the codomain of a surjection is its domain. (Contributed by AV, 29-Sep-2024.)
Assertion
Ref Expression
focnvimacdmdm (𝐺:𝐴onto𝐵 → (𝐺𝐵) = 𝐴)

Proof of Theorem focnvimacdmdm
StepHypRef Expression
1 forn 6760 . . . . 5 (𝐺:𝐴onto𝐵 → ran 𝐺 = 𝐵)
21eqcomd 2743 . . . 4 (𝐺:𝐴onto𝐵𝐵 = ran 𝐺)
32imaeq2d 6014 . . 3 (𝐺:𝐴onto𝐵 → (𝐺𝐵) = (𝐺 “ ran 𝐺))
4 cnvimarndm 6035 . . 3 (𝐺 “ ran 𝐺) = dom 𝐺
53, 4eqtrdi 2793 . 2 (𝐺:𝐴onto𝐵 → (𝐺𝐵) = dom 𝐺)
6 fof 6757 . . 3 (𝐺:𝐴onto𝐵𝐺:𝐴𝐵)
76fdmd 6680 . 2 (𝐺:𝐴onto𝐵 → dom 𝐺 = 𝐴)
85, 7eqtrd 2777 1 (𝐺:𝐴onto𝐵 → (𝐺𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  ccnv 5633  dom cdm 5634  ran crn 5635  cima 5637  ontowfo 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:  foco  6771
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