Theorem imadifssrn 6200

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 3953	. . . . . 6 ⊢ (dom 𝐹 ∖ 𝐴) ⊆ (dom 𝐹 ∖ 𝐴)
2		ssundif 4443	. . . . . 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 6066	. . . . 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 6141	. . . 4 ⊢ (𝐹 “ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))) = ((𝐹 “ 𝐴) ∪ (𝐹 “ (dom 𝐹 ∖ 𝐴)))
7	5, 6	eqtri 2784	. . 3 ⊢ ran 𝐹 = ((𝐹 “ 𝐴) ∪ (𝐹 “ (dom 𝐹 ∖ 𝐴)))
8		df-ima 5664	. . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
9	8	sseq2i 3960	. . . 4 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴) ↔ (𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴))
10		ssequn2 4135	. . . 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 2808	. 2 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = (𝐹 “ 𝐴))
13	12, 8	eqtrdi 2812	1 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ 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:  cyclnumvtx  30381