Theorem focnvimacdmdm 6808

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 6799	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → ran 𝐺 = 𝐵)
2	1	eqcomd 2766	. . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐵 = ran 𝐺)
3	2	imaeq2d 6051	. . 3 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = (◡𝐺 “ ran 𝐺))
4		cnvimarndm 6079	. . 3 ⊢ (◡𝐺 “ ran 𝐺) = dom 𝐺
5	3, 4	eqtrdi 2811	. 2 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = dom 𝐺)
6		fof 6796	. . 3 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐺:𝐴⟶𝐵)
7	6	fdmd 6720	. 2 ⊢ (𝐺:𝐴–onto→𝐵 → dom 𝐺 = 𝐴)
8	5, 7	eqtrd 2795	1 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   “ cima 5653  –onto→wfo 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-fn 6541  df-f 6542  df-fo 6544

This theorem is used by:  foco  6810