Theorem focnvimacdmdm 6765

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 6756	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → ran 𝐺 = 𝐵)
2	1	eqcomd 2743	. . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐵 = ran 𝐺)
3	2	imaeq2d 6026	. . 3 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = (◡𝐺 “ ran 𝐺))
4		cnvimarndm 6049	. . 3 ⊢ (◡𝐺 “ ran 𝐺) = dom 𝐺
5	3, 4	eqtrdi 2788	. 2 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = dom 𝐺)
6		fof 6753	. . 3 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐺:𝐴⟶𝐵)
7	6	fdmd 6679	. 2 ⊢ (𝐺:𝐴–onto→𝐵 → dom 𝐺 = 𝐴)
8	5, 7	eqtrd 2772	1 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = 𝐴)

Syntax hints:   → wi 4   = wceq 1542  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   “ cima 5634  –onto→wfo 6497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fn 6502  df-f 6503  df-fo 6505

This theorem is referenced by:  foco  6767