Theorem imadmrn 2610

Description: The image of the domain of a class is the range of the class.

Assertion

Ref	Expression
imadmrn	⊢ (A “ dom A) = ran A

Proof of Theorem imadmrn

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . 7 ⊢ x ∈ V
2	1	opeldm 2534	. . . . . 6 ⊢ (⟨x, y⟩ ∈ A → x ∈ dom A)
3	2	pm4.71i 483	. . . . 5 ⊢ (⟨x, y⟩ ∈ A ↔ (⟨x, y⟩ ∈ A ∧ x ∈ dom A))
4		ancom 333	. . . . 5 ⊢ ((⟨x, y⟩ ∈ A ∧ x ∈ dom A) ↔ (x ∈ dom A ∧ ⟨x, y⟩ ∈ A))
5	3, 4	bitr2 152	. . . 4 ⊢ ((x ∈ dom A ∧ ⟨x, y⟩ ∈ A) ↔ ⟨x, y⟩ ∈ A)
6	5	biex 733	. . 3 ⊢ (∃x(x ∈ dom A ∧ ⟨x, y⟩ ∈ A) ↔ ∃x⟨x, y⟩ ∈ A)
7	6	biabi 1181	. 2 ⊢ {y∣∃x(x ∈ dom A ∧ ⟨x, y⟩ ∈ A)} = {y∣∃x⟨x, y⟩ ∈ A}
8		dfima3 2605	. 2 ⊢ (A “ dom A) = {y∣∃x(x ∈ dom A ∧ ⟨x, y⟩ ∈ A)}
9		dfrn3 2524	. 2 ⊢ ran A = {y∣∃x⟨x, y⟩ ∈ A}
10	7, 8, 9	3eqtr4 1126	1 ⊢ (A “ dom A) = ran A

Syntax hints:   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  dom cdm 2410  ran crn 2411   “ cima 2413

This theorem is referenced by:  fnima 2738  fnex 2740  foima 2790  f1imacnv 2814  fsn2 2896  mapsn 3269  phplem5 3407  php3 3411  unir1 3511

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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