Theorem imadifssran 6202

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 5674	. . 3 ⊢ (𝐹 “ (dom 𝐹 ∖ 𝐴)) = ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))
2	1	sseq1i 3964	. 2 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) ↔ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴))
3		ssun2 4131	. . . . . . . . . . . . 13 ⊢ dom 𝐹 ⊆ (𝐴 ∪ dom 𝐹)
4		undif2 4437	. . . . . . . . . . . . 13 ⊢ (𝐴 ∪ (dom 𝐹 ∖ 𝐴)) = (𝐴 ∪ dom 𝐹)
5	3, 4	sseqtrri 3985	. . . . . . . . . . . 12 ⊢ dom 𝐹 ⊆ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))
6		ssres2 6003	. . . . . . . . . . . 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 5992	. . . . . . . . . . 11 ⊢ (𝐹 ↾ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))) = ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
9	7, 8	sseqtri 3984	. . . . . . . . . 10 ⊢ (𝐹 ↾ dom 𝐹) ⊆ ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
10	9	rnssi 5930	. . . . . . . . 9 ⊢ ran (𝐹 ↾ dom 𝐹) ⊆ ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
11		rnun 6142	. . . . . . . . 9 ⊢ ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴))) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
12	10, 11	sseqtri 3984	. . . . . . . 8 ⊢ ran (𝐹 ↾ dom 𝐹) ⊆ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
13	12	sseli 3932	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝐹 ↾ dom 𝐹) → 𝑦 ∈ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
14		elun 4106	. . . . . . 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 4354	. . . . . . . . . . 11 ⊢ (dom 𝐹 ∩ V) = dom 𝐹
17	16	ineqcomi 4163	. . . . . . . . . 10 ⊢ (V ∩ dom 𝐹) = dom 𝐹
18	17	reseq2i 5975	. . . . . . . . 9 ⊢ (𝐹 ↾ (V ∩ dom 𝐹)) = (𝐹 ↾ dom 𝐹)
19		resindm 6029	. . . . . . . . 9 ⊢ (𝐹 ↾ (V ∩ dom 𝐹)) = (𝐹 ↾ V)
20	18, 19	eqtr3i 2786	. . . . . . . 8 ⊢ (𝐹 ↾ dom 𝐹) = (𝐹 ↾ V)
21	20	rneqi 5927	. . . . . . 7 ⊢ ran (𝐹 ↾ dom 𝐹) = ran (𝐹 ↾ V)
22		rnresv 6200	. . . . . . 7 ⊢ ran (𝐹 ↾ V) = ran 𝐹
23	21, 22	eqtr2i 2785	. . . . . 6 ⊢ ran 𝐹 = ran (𝐹 ↾ dom 𝐹)
24	15, 23	eleq2s 2879	. . . . 5 ⊢ (𝑦 ∈ ran 𝐹 → (𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
25		ssel 3930	. . . . 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 871	. . . . 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 3942	. . 3 ⊢ (ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 ⊆ ran (𝐹 ↾ 𝐴))
30		rnresss 6016	. . . 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 3953	. 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))

Syntax hints:   → wi 4   ∨ wo 860   = wceq 1568   ∈ wcel 2141  Vcvv 3453   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  dom cdm 5661  ran crn 5662   ↾ cres 5663   “ cima 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674

This theorem is referenced by:  cyclnumvtx  30115