Theorem imadifssranOLD 6203

Description: Obsolete version of imadifssran 6202 as of 10-Jul-2026. (Contributed by AV, 4-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
imadifssranOLD	⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴)))

Proof of Theorem imadifssranOLD

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 3965	. 2 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) ↔ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴))
3		ssel 3931	. . . . . . 7 ⊢ (ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → (𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)))
4		resdm 6025	. . . . . . . . . . . . . . 15 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
5	4	eqcomd 2769	. . . . . . . . . . . . . 14 ⊢ (Rel 𝐹 → 𝐹 = (𝐹 ↾ dom 𝐹))
6	5	adantr 485	. . . . . . . . . . . . 13 ⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → 𝐹 = (𝐹 ↾ dom 𝐹))
7	6	rneqd 5928	. . . . . . . . . . . 12 ⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ran 𝐹 = ran (𝐹 ↾ dom 𝐹))
8	7	eleq2d 2849	. . . . . . . . . . 11 ⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ ran (𝐹 ↾ dom 𝐹)))
9		undif 4443	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ dom 𝐹 ↔ (𝐴 ∪ (dom 𝐹 ∖ 𝐴)) = dom 𝐹)
10	9	biimpi 219	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ⊆ dom 𝐹 → (𝐴 ∪ (dom 𝐹 ∖ 𝐴)) = dom 𝐹)
11	10	eqcomd 2769	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ⊆ dom 𝐹 → dom 𝐹 = (𝐴 ∪ (dom 𝐹 ∖ 𝐴)))
12	11	reseq2d 5978	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ⊆ dom 𝐹 → (𝐹 ↾ dom 𝐹) = (𝐹 ↾ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))))
13		resundi 5992	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ↾ (𝐴 ∪ (dom 𝐹 ∖ 𝐴))) = ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
14	12, 13	eqtrdi 2814	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ dom 𝐹 → (𝐹 ↾ dom 𝐹) = ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
15	14	rneqd 5928	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ dom 𝐹 → ran (𝐹 ↾ dom 𝐹) = ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
16		rnun 6142	. . . . . . . . . . . . . . 15 ⊢ ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ (dom 𝐹 ∖ 𝐴))) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))
17	15, 16	eqtrdi 2814	. . . . . . . . . . . . . 14 ⊢ (𝐴 ⊆ dom 𝐹 → ran (𝐹 ↾ dom 𝐹) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
18	17	eleq2d 2849	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑦 ∈ ran (𝐹 ↾ dom 𝐹) ↔ 𝑦 ∈ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))))
19		elun 4107	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))) ↔ (𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴))))
20	18, 19	bitrdi 290	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑦 ∈ ran (𝐹 ↾ dom 𝐹) ↔ (𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))))
21	20	adantl 486	. . . . . . . . . . 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 282	. . . . . . . . . 10 ⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑦 ∈ ran 𝐹 ↔ (𝑦 ∈ ran (𝐹 ↾ 𝐴) ∨ 𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)))))
23	22	adantl 486	. . . . . . . . 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 43	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → ((𝑦 ∈ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)) → 𝑦 ∈ ran (𝐹 ↾ 𝐴)))
25	24	jao1i 871	. . . . . . . . . . 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 33	. . . . . . . . . 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 485	. . . . . . . . 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 243	. . . . . . . 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 417	. . . . . . 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 18	. . . . . 6 ⊢ (ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ ran (𝐹 ↾ 𝐴))))
31	30	impcom 412	. . . . 5 ⊢ (((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴)) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ ran (𝐹 ↾ 𝐴)))
32	31	ssrdv 3943	. . . 4 ⊢ (((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴)) → ran 𝐹 ⊆ ran (𝐹 ↾ 𝐴))
33		rnresss 6016	. . . . 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 3954	. . 3 ⊢ (((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴)) → ran 𝐹 = ran (𝐹 ↾ 𝐴))
36	35	ex 417	. 2 ⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (ran (𝐹 ↾ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴)))
37	2, 36	biimtrid 245	1 ⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ∖ cdif 3902   ∪ cun 3903   ⊆ wss 3905  dom cdm 5661  ran crn 5662   ↾ cres 5663   “ cima 5664  Rel wrel 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  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: (None)