Theorem focnvimacdmdm 6830

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

Assertion

Ref	Expression
focnvimacdmdm	⊢ (𝐺:𝐴⟶onto𝐵 → (◡𝐺 “ 𝐵) = 𝐴)

Proof of Theorem focnvimacdmdm

Step	Hyp	Ref	Expression
1		forn 6821	. . . . 5 ⊢ (𝐺:𝐴⟶onto𝐵 → ran 𝐺 = 𝐵)
2	1	eqcomd 2769	. . . 4 ⊢ (𝐺:𝐴⟶onto𝐵 → 𝐵 = ran 𝐺)
3	2	imaeq2d 6064	. . 3 ⊢ (𝐺:𝐴⟶onto𝐵 → (◡𝐺 “ 𝐵) = (◡𝐺 “ ran 𝐺))
4		cnvimarndm 6092	. . 3 ⊢ (◡𝐺 “ ran 𝐺) = dom 𝐺
5	3, 4	eqtrdi 2814	. 2 ⊢ (𝐺:𝐴⟶onto𝐵 → (◡𝐺 “ 𝐵) = dom 𝐺)
6		fof 6818	. . 3 ⊢ (𝐺:𝐴⟶onto𝐵 → 𝐺:𝐴⟶𝐵)
7	6	fdmd 6742	. 2 ⊢ (𝐺:𝐴⟶onto𝐵 → dom 𝐺 = 𝐴)
8	5, 7	eqtrd 2798	1 ⊢ (𝐺:𝐴⟶onto𝐵 → (◡𝐺 “ 𝐵) = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658  ⟶ontowfo 6556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fn 6561  df-f 6562  df-fo 6564

This theorem is used by:  foco  6832