Theorem imadifssrn 6201

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 3930	. . . . . 6 ⊢ (dom 𝐹 ∖ 𝐴) ⊆ (dom 𝐹 ∖ 𝐴)
2		ssundif 4442	. . . . . 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 6065	. . . . 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 6140	. . . 4 ⊢ (𝐹 “ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))) = ((𝐹 “ 𝐴) ∪ (𝐹 “ (dom 𝐹 ∖ 𝐴)))
7	5, 6	eqtri 2783	. . 3 ⊢ ran 𝐹 = ((𝐹 “ 𝐴) ∪ (𝐹 “ (dom 𝐹 ∖ 𝐴)))
8		df-ima 5663	. . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
9	8	sseq2i 3937	. . . 4 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴) ↔ (𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴))
10		ssequn2 4187	. . . 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 2807	. 2 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = (𝐹 “ 𝐴))
13	12, 8	eqtrdi 2811	1 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ⊆ wss 3895   ∖ cdif 4045   ∪ cun 4149  dom cdm 5650  ran crn 5651   ↾ cres 5652   “ 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:  cyclnumvtx  30394