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Theorem focnvimacdmdm 6816
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 6807 . . . . 5 (𝐺:𝐴onto𝐵 → ran 𝐺 = 𝐵)
21eqcomd 2731 . . . 4 (𝐺:𝐴onto𝐵𝐵 = ran 𝐺)
32imaeq2d 6059 . . 3 (𝐺:𝐴onto𝐵 → (𝐺𝐵) = (𝐺 “ ran 𝐺))
4 cnvimarndm 6082 . . 3 (𝐺 “ ran 𝐺) = dom 𝐺
53, 4eqtrdi 2781 . 2 (𝐺:𝐴onto𝐵 → (𝐺𝐵) = dom 𝐺)
6 fof 6804 . . 3 (𝐺:𝐴onto𝐵𝐺:𝐴𝐵)
76fdmd 6727 . 2 (𝐺:𝐴onto𝐵 → dom 𝐺 = 𝐴)
85, 7eqtrd 2765 1 (𝐺:𝐴onto𝐵 → (𝐺𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  ccnv 5672  dom cdm 5673  ran crn 5674  cima 5676  ontowfo 6541
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 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
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 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5145  df-opab 5207  df-xp 5679  df-cnv 5681  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-fn 6546  df-f 6547  df-fo 6549
This theorem is referenced by:  foco  6818
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