Theorem focnvimacdmdm 6801

Description: The preimage of the codomain of a surjection is its domain. (Contributed by AV, 29-Sep-2024.)

Assertion

Ref	Expression
focnvimacdmdm	⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = 𝐴)

Proof of Theorem focnvimacdmdm

Step	Hyp	Ref	Expression
1		forn 6792	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → ran 𝐺 = 𝐵)
2	1	eqcomd 2741	. . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐵 = ran 𝐺)
3	2	imaeq2d 6047	. . 3 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = (◡𝐺 “ ran 𝐺))
4		cnvimarndm 6070	. . 3 ⊢ (◡𝐺 “ ran 𝐺) = dom 𝐺
5	3, 4	eqtrdi 2786	. 2 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = dom 𝐺)
6		fof 6789	. . 3 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐺:𝐴⟶𝐵)
7	6	fdmd 6715	. 2 ⊢ (𝐺:𝐴–onto→𝐵 → dom 𝐺 = 𝐴)
8	5, 7	eqtrd 2770	1 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = 𝐴)

Syntax hints:   → wi 4   = wceq 1540  ^◡ccnv 5653  dom cdm 5654  ran crn 5655   “ cima 5657  –onto→wfo 6528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-fn 6533  df-f 6534  df-fo 6536

This theorem is referenced by:  foco  6803