Theorem dfrn7 6065

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 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	opeldm 5888	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
4		ssel 3902	. . . . . 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 1953	. . 3 ⊢ (dom 𝐴 ⊆ 𝐵 → (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
8	7	abbidv 2826	. 2 ⊢ (dom 𝐴 ⊆ 𝐵 → {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)})
9		dfrn3 5870	. 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
10		dfima3 6058	. 2 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
11	8, 9, 10	3eqtr4g 2820	1 ⊢ (dom 𝐴 ⊆ 𝐵 → ran 𝐴 = (𝐴 “ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738   ⊆ wss 3895  ⟨cop 4834  dom cdm 5650  ran crn 5651   “ cima 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663

This theorem is used by:  imadmrn  6066  cnvimassrndm  6078  imadifssrn  6201  rnr1  9778