Theorem dfrn7 6084

Description: The range is equal to the image of any class including the domain. (Contributed by BJ, 27-Sep-2026.)

Assertion

Ref	Expression
dfrn7	⊢ (dom 𝐴 ⊆ 𝐵 → ran 𝐴 = (𝐴 “ 𝐵))

Proof of Theorem dfrn7

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 3458	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	opeldm 5901	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
4		ssel 3907	. . . . . 6 ⊢ (dom 𝐴 ⊆ 𝐵 → (𝑥 ∈ dom 𝐴 → 𝑥 ∈ 𝐵))
5	3, 4	syl5 35	. . . . 5 ⊢ (dom 𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ 𝐵))
6	5	pm4.71rd 572	. . . 4 ⊢ (dom 𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
7	6	exbidv 1956	. . 3 ⊢ (dom 𝐴 ⊆ 𝐵 → (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
8	7	abbidv 2829	. 2 ⊢ (dom 𝐴 ⊆ 𝐵 → {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)})
9		dfrn3 5883	. 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
10		dfima3 6077	. 2 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
11	8, 9, 10	3eqtr4g 2823	1 ⊢ (dom 𝐴 ⊆ 𝐵 → ran 𝐴 = (𝐴 “ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741   ⊆ wss 3900  ⟨cop 4840  dom cdm 5658  ran crn 5659   “ cima 5661

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 5255  ax-pr 5398

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 4841  df-br 5109  df-opab 5174  df-xp 5664  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671

This theorem is used by:  imadmrn  6085  cnvimassrndm  6097  imadifssrn  6220  rnr1  9811