Theorem imadifssran 6110

Description: Condition for the range of a relation to be the range of one its restrictions. (Contributed by AV, 4-Oct-2025.)

Assertion

Ref	Expression
imadifssran	⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((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 5638	. . 3 ⊢ (𝐹 “ (dom 𝐹 ∖ 𝐴)) = ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))
2	1	sseq1i 3963	. 2 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) ↔ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴))
3		ssel 3928	. . . . . . 7 ⊢ (ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → (𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)))
4		resdm 5986	. . . . . . . . . . . . . . 15 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
5	4	eqcomd 2743	. . . . . . . . . . . . . 14 ⊢ (Rel 𝐹 → 𝐹 = (𝐹 ↾ dom 𝐹))
6	5	adantr 480	. . . . . . . . . . . . 13 ⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → 𝐹 = (𝐹 ↾ dom 𝐹))
7	6	rneqd 5888	. . . . . . . . . . . 12 ⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ran 𝐹 = ran (𝐹 ↾ dom 𝐹))
8	7	eleq2d 2823	. . . . . . . . . . 11 ⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ ran (𝐹 ↾ dom 𝐹)))
9		undif 4435	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ dom 𝐹 ↔ (𝐴 ∪ (dom 𝐹 ∖ 𝐴)) = dom 𝐹)
10	9	biimpi 216	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ⊆ dom 𝐹 → (𝐴 ∪ (dom 𝐹 ∖ 𝐴)) = dom 𝐹)
11	10	eqcomd 2743	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ⊆ dom 𝐹 → dom 𝐹 = (𝐴 ∪ (dom 𝐹 ∖ 𝐴)))
12	11	reseq2d 5939	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ⊆ dom 𝐹 → (𝐹 ↾ dom 𝐹) = (𝐹 ↾ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))))
13		resundi 5953	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ↾ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))) = ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
14	12, 13	eqtrdi 2788	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ dom 𝐹 → (𝐹 ↾ dom 𝐹) = ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
15	14	rneqd 5888	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ dom 𝐹 → ran (𝐹 ↾ dom 𝐹) = ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
16		rnun 6104	. . . . . . . . . . . . . . 15 ⊢ ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴))) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
17	15, 16	eqtrdi 2788	. . . . . . . . . . . . . 14 ⊢ (𝐴 ⊆ dom 𝐹 → ran (𝐹 ↾ dom 𝐹) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
18	17	eleq2d 2823	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑦 ∈ ran (𝐹 ↾ dom 𝐹) ↔ 𝑦 ∈ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))))
19		elun 4106	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))) ↔ (𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
20	18, 19	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑦 ∈ ran (𝐹 ↾ dom 𝐹) ↔ (𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))))
21	20	adantl 481	. . . . . . . . . . 11 ⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑦 ∈ ran (𝐹 ↾ dom 𝐹) ↔ (𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))))
22	8, 21	bitrd 279	. . . . . . . . . 10 ⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑦 ∈ ran 𝐹 ↔ (𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))))
23	22	adantl 481	. . . . . . . . 9 ⊢ (((𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)) ∧ (Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹)) → (𝑦 ∈ ran 𝐹 ↔ (𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))))
24		pm2.27 42	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → ((𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)))
25	24	jao1i 859	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))) → ((𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)))
26	25	com12 32	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)) → ((𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)))
27	26	adantr 480	. . . . . . . . 9 ⊢ (((𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)) ∧ (Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹)) → ((𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)))
28	23, 27	sylbid 240	. . . . . . . 8 ⊢ (((𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)) ∧ (Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹)) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ ran (𝐹 ↾ 𝐴)))
29	28	ex 412	. . . . . . 7 ⊢ ((𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)) → ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ ran (𝐹 ↾ 𝐴))))
30	3, 29	syl 17	. . . . . 6 ⊢ (ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ ran (𝐹 ↾ 𝐴))))
31	30	impcom 407	. . . . 5 ⊢ (((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴)) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ ran (𝐹 ↾ 𝐴)))
32	31	ssrdv 3940	. . . 4 ⊢ (((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴)) → ran 𝐹 ⊆ ran (𝐹 ↾ 𝐴))
33		rnresss 5977	. . . . 5 ⊢ ran (𝐹 ↾ 𝐴) ⊆ ran 𝐹
34	33	a1i 11	. . . 4 ⊢ (((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴)) → ran (𝐹 ↾ 𝐴) ⊆ ran 𝐹)
35	32, 34	eqssd 3952	. . 3 ⊢ (((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴)) → ran 𝐹 = ran (𝐹 ↾ 𝐴))
36	35	ex 412	. 2 ⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴)))
37	2, 36	biimtrid 242	1 ⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ∖ cdif 3899   ∪ cun 3900   ⊆ wss 3902  dom cdm 5625  ran crn 5626   ↾ cres 5627   “ cima 5628  Rel wrel 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638

This theorem is referenced by:  cyclnumvtx  29856