Theorem imadmrn 3428

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 1820	. . . . . . 7 ⊢ x ∈ V
2	1	opeldm 3328	. . . . . 6 ⊢ (⟨x, y⟩ ∈ A → x ∈ dom A)
3	2	pm4.71i 640	. . . . 5 ⊢ (⟨x, y⟩ ∈ A ↔ (⟨x, y⟩ ∈ A ⋀ x ∈ dom A))
4		ancom 438	. . . . 5 ⊢ ((⟨x, y⟩ ∈ A ⋀ x ∈ dom A) ↔ (x ∈ dom A ⋀ ⟨x, y⟩ ∈ A))
5	3, 4	bitr2 174	. . . 4 ⊢ ((x ∈ dom A ⋀ ⟨x, y⟩ ∈ A) ↔ ⟨x, y⟩ ∈ A)
6	5	exbii 1057	. . 3 ⊢ (∃x(x ∈ dom A ⋀ ⟨x, y⟩ ∈ A) ↔ ∃x⟨x, y⟩ ∈ A)
7	6	abbii 1582	. 2 ⊢ {y∣∃x(x ∈ dom A ⋀ ⟨x, y⟩ ∈ A)} = {y∣∃x⟨x, y⟩ ∈ A}
8		dfima3 3420	. 2 ⊢ (A “ dom A) = {y∣∃x(x ∈ dom A ⋀ ⟨x, y⟩ ∈ A)}
9		dfrn3 3318	. 2 ⊢ ran A = {y∣∃x⟨x, y⟩ ∈ A}
10	7, 8, 9	3eqtr4 1512	1 ⊢ (A “ dom A) = ran A

Syntax hints:   ⋀ wa 223   = wceq 960   ∈ wcel 962  ∃wex 984  {cab 1470  ⟨cop 2421  dom cdm 3184  ran crn 3185   “ cima 3187

This theorem is referenced by:  fnima 3618  fnex 3621  foima 3690  f1imacnv 3719  fsn2 3850  elunirn 3882  mapsn 4359  phplem4 4526  php3 4530  unir1 4679  cnconst 7789

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-sep 2716  ax-pow 2756  ax-pr 2793

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-rex 1657  df-v 1819  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-pw 2412  df-sn 2422  df-pr 2423  df-op 2426  df-br 2633  df-opab 2680  df-xp 3198  df-cnv 3200  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205

Copyright terms: Public domain