Theorem imadifssran 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.)

Assertion

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

Proof of Theorem imadifssran

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ima 5672	. . 3 ⊢ (𝐹 “ (dom 𝐹 ∖ 𝐴)) = ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))
2	1	sseq1i 3962	. 2 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) ↔ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴))
3		ssun2 4128	. . . . . . . . . . . . 13 ⊢ dom 𝐹 ⊆ (𝐴 ∪ dom 𝐹)
4		undif2 4434	. . . . . . . . . . . . 13 ⊢ (𝐴 ∪ (dom 𝐹 ∖ 𝐴)) = (𝐴 ∪ dom 𝐹)
5	3, 4	sseqtrri 3983	. . . . . . . . . . . 12 ⊢ dom 𝐹 ⊆ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))
6		ssres2 6001	. . . . . . . . . . . 12 ⊢ (dom 𝐹 ⊆ (𝐴 ∪ (dom 𝐹 ∖ 𝐴)) → (𝐹 ↾ dom 𝐹) ⊆ (𝐹 ↾ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))))
7	5, 6	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐹 ↾ dom 𝐹) ⊆ (𝐹 ↾ (𝐴 ∪ (dom 𝐹 ∖ 𝐴)))
8		resundi 5990	. . . . . . . . . . 11 ⊢ (𝐹 ↾ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))) = ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
9	7, 8	sseqtri 3982	. . . . . . . . . 10 ⊢ (𝐹 ↾ dom 𝐹) ⊆ ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
10	9	rnssi 5928	. . . . . . . . 9 ⊢ ran (𝐹 ↾ dom 𝐹) ⊆ ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
11		rnun 6140	. . . . . . . . 9 ⊢ ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴))) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
12	10, 11	sseqtri 3982	. . . . . . . 8 ⊢ ran (𝐹 ↾ dom 𝐹) ⊆ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
13	12	sseli 3930	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝐹 ↾ dom 𝐹) → 𝑦 ∈ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
14		elun 4103	. . . . . . 7 ⊢ (𝑦 ∈ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))) ↔ (𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
15	13, 14	sylib 221	. . . . . 6 ⊢ (𝑦 ∈ ran (𝐹 ↾ dom 𝐹) → (𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
16		inv1 4351	. . . . . . . . . . 11 ⊢ (dom 𝐹 ∩ V) = dom 𝐹
17	16	ineqcomi 4160	. . . . . . . . . 10 ⊢ (V ∩ dom 𝐹) = dom 𝐹
18	17	reseq2i 5973	. . . . . . . . 9 ⊢ (𝐹 ↾ (V ∩ dom 𝐹)) = (𝐹 ↾ dom 𝐹)
19		resindm 6027	. . . . . . . . 9 ⊢ (𝐹 ↾ (V ∩ dom 𝐹)) = (𝐹 ↾ V)
20	18, 19	eqtr3i 2787	. . . . . . . 8 ⊢ (𝐹 ↾ dom 𝐹) = (𝐹 ↾ V)
21	20	rneqi 5925	. . . . . . 7 ⊢ ran (𝐹 ↾ dom 𝐹) = ran (𝐹 ↾ V)
22		rnresv 6199	. . . . . . 7 ⊢ ran (𝐹 ↾ V) = ran 𝐹
23	21, 22	eqtr2i 2786	. . . . . 6 ⊢ ran 𝐹 = ran (𝐹 ↾ dom 𝐹)
24	15, 23	eleq2s 2880	. . . . 5 ⊢ (𝑦 ∈ ran 𝐹 → (𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
25		ssel 3928	. . . . 5 ⊢ (ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → (𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)))
26		pm2.27 43	. . . . . 6 ⊢ (𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → ((𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)))
27	26	jao1i 872	. . . . 5 ⊢ ((𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))) → ((𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)))
28	24, 25, 27	syl2imc 42	. . . 4 ⊢ (ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ ran (𝐹 ↾ 𝐴)))
29	28	ssrdv 3940	. . 3 ⊢ (ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 ⊆ ran (𝐹 ↾ 𝐴))
30		rnresss 6014	. . . 4 ⊢ ran (𝐹 ↾ 𝐴) ⊆ ran 𝐹
31	30	a1i 11	. . 3 ⊢ (ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran (𝐹 ↾ 𝐴) ⊆ ran 𝐹)
32	29, 31	eqssd 3951	. 2 ⊢ (ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴))
33	2, 32	sylbi 220	1 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  dom cdm 5659  ran crn 5660   ↾ cres 5661   “ cima 5662

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 2734  ax-sep 5255  ax-pr 5402

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672

This theorem is used by:  cyclnumvtx  30268