Theorem focnvimacdmdm 6790

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 6781	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → ran 𝐺 = 𝐵)
2	1	eqcomd 2769	. . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐵 = ran 𝐺)
3	2	imaeq2d 6049	. . 3 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = (◡𝐺 “ ran 𝐺))
4		cnvimarndm 6072	. . 3 ⊢ (◡𝐺 “ ran 𝐺) = dom 𝐺
5	3, 4	eqtrdi 2814	. 2 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = dom 𝐺)
6		fof 6778	. . 3 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐺:𝐴⟶𝐵)
7	6	fdmd 6702	. 2 ⊢ (𝐺:𝐴–onto→𝐵 → dom 𝐺 = 𝐴)
8	5, 7	eqtrd 2798	1 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = 𝐴)

Syntax hints:   → wi 4   = wceq 1561  ^◡ccnv 5647  dom cdm 5648  ran crn 5649   “ cima 5651  –onto→wfo 6519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-xp 5654  df-cnv 5656  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-fn 6524  df-f 6525  df-fo 6527

This theorem is referenced by:  foco  6792