Theorem imadifssrn 6220

Description: Condition for the range of a class to be the range of one of its restrictions. (Contributed by AV, 4-Oct-2025.) Remove antecedent. (Revised by Eric Schmidt, 10-Jul-2026.) (Proof shortened by BJ, 29-Sep-2026.)

Assertion

Ref	Expression
imadifssrn	⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴))

Proof of Theorem imadifssrn

Step	Hyp	Ref	Expression
1		ssid 3935	. . . . . 6 ⊢ (dom 𝐹 ∖ 𝐴) ⊆ (dom 𝐹 ∖ 𝐴)
2		ssundif 4446	. . . . . 6 ⊢ (dom 𝐹 ⊆ (𝐴 ∪ (dom 𝐹 ∖ 𝐴)) ↔ (dom 𝐹 ∖ 𝐴) ⊆ (dom 𝐹 ∖ 𝐴))
3	1, 2	mpbir 234	. . . . 5 ⊢ dom 𝐹 ⊆ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))
4		dfrn7 6084	. . . . 5 ⊢ (dom 𝐹 ⊆ (𝐴 ∪ (dom 𝐹 ∖ 𝐴)) → ran 𝐹 = (𝐹 “ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))))
5	3, 4	ax-mp 5	. . . 4 ⊢ ran 𝐹 = (𝐹 “ (𝐴 ∪ (dom 𝐹 ∖ 𝐴)))
6		imaundi 6159	. . . 4 ⊢ (𝐹 “ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))) = ((𝐹 “ 𝐴) ∪ (𝐹 “ (dom 𝐹 ∖ 𝐴)))
7	5, 6	eqtri 2786	. . 3 ⊢ ran 𝐹 = ((𝐹 “ 𝐴) ∪ (𝐹 “ (dom 𝐹 ∖ 𝐴)))
8		df-ima 5671	. . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
9	8	sseq2i 3942	. . . 4 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴) ↔ (𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴))
10		ssequn2 4190	. . . 4 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴) ↔ ((𝐹 “ 𝐴) ∪ (𝐹 “ (dom 𝐹 ∖ 𝐴))) = (𝐹 “ 𝐴))
11	9, 10	sylbb1 240	. . 3 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ((𝐹 “ 𝐴) ∪ (𝐹 “ (dom 𝐹 ∖ 𝐴))) = (𝐹 “ 𝐴))
12	7, 11	eqtrid 2810	. 2 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = (𝐹 “ 𝐴))
13	12, 8	eqtrdi 2814	1 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3900   ∖ cdif 4048   ∪ cun 4152  dom cdm 5658  ran crn 5659   ↾ cres 5660   “ 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:  cyclnumvtx  30445