Theorem imadifssran 6191

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 5660	. . 3 ⊢ (𝐹 “ (dom 𝐹 ∖ 𝐴)) = ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))
2	1	sseq1i 3958	. 2 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) ↔ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴))
3		ssun2 4124	. . . . . . . . . . . . 13 ⊢ dom 𝐹 ⊆ (𝐴 ∪ dom 𝐹)
4		undif2 4430	. . . . . . . . . . . . 13 ⊢ (𝐴 ∪ (dom 𝐹 ∖ 𝐴)) = (𝐴 ∪ dom 𝐹)
5	3, 4	sseqtrri 3979	. . . . . . . . . . . 12 ⊢ dom 𝐹 ⊆ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))
6		ssres2 5991	. . . . . . . . . . . 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 5980	. . . . . . . . . . 11 ⊢ (𝐹 ↾ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))) = ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
9	7, 8	sseqtri 3978	. . . . . . . . . 10 ⊢ (𝐹 ↾ dom 𝐹) ⊆ ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
10	9	rnssi 5918	. . . . . . . . 9 ⊢ ran (𝐹 ↾ dom 𝐹) ⊆ ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
11		rnun 6130	. . . . . . . . 9 ⊢ ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴))) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
12	10, 11	sseqtri 3978	. . . . . . . 8 ⊢ ran (𝐹 ↾ dom 𝐹) ⊆ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
13	12	sseli 3926	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝐹 ↾ dom 𝐹) → 𝑦 ∈ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
14		elun 4099	. . . . . . 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 4347	. . . . . . . . . . 11 ⊢ (dom 𝐹 ∩ V) = dom 𝐹
17	16	ineqcomi 4156	. . . . . . . . . 10 ⊢ (V ∩ dom 𝐹) = dom 𝐹
18	17	reseq2i 5963	. . . . . . . . 9 ⊢ (𝐹 ↾ (V ∩ dom 𝐹)) = (𝐹 ↾ dom 𝐹)
19		resindm 6017	. . . . . . . . 9 ⊢ (𝐹 ↾ (V ∩ dom 𝐹)) = (𝐹 ↾ V)
20	18, 19	eqtr3i 2785	. . . . . . . 8 ⊢ (𝐹 ↾ dom 𝐹) = (𝐹 ↾ V)
21	20	rneqi 5915	. . . . . . 7 ⊢ ran (𝐹 ↾ dom 𝐹) = ran (𝐹 ↾ V)
22		rnresv 6189	. . . . . . 7 ⊢ ran (𝐹 ↾ V) = ran 𝐹
23	21, 22	eqtr2i 2784	. . . . . 6 ⊢ ran 𝐹 = ran (𝐹 ↾ dom 𝐹)
24	15, 23	eleq2s 2878	. . . . 5 ⊢ (𝑦 ∈ ran 𝐹 → (𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
25		ssel 3924	. . . . 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 3936	. . 3 ⊢ (ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 ⊆ ran (𝐹 ↾ 𝐴))
30		rnresss 6004	. . . 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 3947	. 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 3450   ∖ cdif 3895   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898  dom cdm 5647  ran crn 5648   ↾ cres 5649   “ cima 5650

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 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 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660

This theorem is used by:  cyclnumvtx  30322