Theorem dfrn7 6066

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 3455	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 3455	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	opeldm 5889	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
4		ssel 3925	. . . . . 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 1954	. . 3 ⊢ (dom 𝐴 ⊆ 𝐵 → (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
8	7	abbidv 2827	. 2 ⊢ (dom 𝐴 ⊆ 𝐵 → {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)})
9		dfrn3 5871	. 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
10		dfima3 6059	. 2 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
11	8, 9, 10	3eqtr4g 2821	1 ⊢ (dom 𝐴 ⊆ 𝐵 → ran 𝐴 = (𝐴 “ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739   ⊆ wss 3899  ⟨cop 4590  dom cdm 5651  ran crn 5652   “ cima 5654

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664

This theorem is used by:  imadmrn  6067  cnvimassrndm  6079  imadifssrn  6200  rnr1  9767