Theorem f1imadifssran 6584

Description: Condition for the range of a one-to-one function to be the range of one its restrictions. Variant of imadifssran 6115. (Contributed by AV, 4-Oct-2025.)

Assertion

Ref	Expression
f1imadifssran	⊢ (Fun ◡𝐹 → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴)))

Proof of Theorem f1imadifssran

Step	Hyp	Ref	Expression
1		imadmrn 6035	. . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		imadif 6582	. . . . . . 7 ⊢ (Fun ◡𝐹 → (𝐹 “ (dom 𝐹 ∖ 𝐴)) = ((𝐹 “ dom 𝐹) ∖ (𝐹 “ 𝐴)))
3	2	sseq1d 3953	. . . . . 6 ⊢ (Fun ◡𝐹 → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴) ↔ ((𝐹 “ dom 𝐹) ∖ (𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐴)))
4		ssundif 4427	. . . . . . 7 ⊢ ((𝐹 “ dom 𝐹) ⊆ ((𝐹 “ 𝐴) ∪ (𝐹 “ 𝐴)) ↔ ((𝐹 “ dom 𝐹) ∖ (𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐴))
5		unidm 4097	. . . . . . . . 9 ⊢ ((𝐹 “ 𝐴) ∪ (𝐹 “ 𝐴)) = (𝐹 “ 𝐴)
6	5	sseq2i 3951	. . . . . . . 8 ⊢ ((𝐹 “ dom 𝐹) ⊆ ((𝐹 “ 𝐴) ∪ (𝐹 “ 𝐴)) ↔ (𝐹 “ dom 𝐹) ⊆ (𝐹 “ 𝐴))
7		id 22	. . . . . . . . 9 ⊢ ((𝐹 “ dom 𝐹) ⊆ (𝐹 “ 𝐴) → (𝐹 “ dom 𝐹) ⊆ (𝐹 “ 𝐴))
8		imassrn 6036	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
9	8, 1	sseqtrri 3971	. . . . . . . . . 10 ⊢ (𝐹 “ 𝐴) ⊆ (𝐹 “ dom 𝐹)
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 “ dom 𝐹) ⊆ (𝐹 “ 𝐴) → (𝐹 “ 𝐴) ⊆ (𝐹 “ dom 𝐹))
11	7, 10	eqssd 3939	. . . . . . . 8 ⊢ ((𝐹 “ dom 𝐹) ⊆ (𝐹 “ 𝐴) → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
12	6, 11	sylbi 217	. . . . . . 7 ⊢ ((𝐹 “ dom 𝐹) ⊆ ((𝐹 “ 𝐴) ∪ (𝐹 “ 𝐴)) → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
13	4, 12	sylbir 235	. . . . . 6 ⊢ (((𝐹 “ dom 𝐹) ∖ (𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐴) → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
14	3, 13	biimtrdi 253	. . . . 5 ⊢ (Fun ◡𝐹 → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴) → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴)))
15	14	imp 406	. . . 4 ⊢ ((Fun ◡𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴)) → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
16	1, 15	eqtr3id 2785	. . 3 ⊢ ((Fun ◡𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴)) → ran 𝐹 = (𝐹 “ 𝐴))
17	16	ex 412	. 2 ⊢ (Fun ◡𝐹 → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴) → ran 𝐹 = (𝐹 “ 𝐴)))
18		df-ima 5644	. . . 4 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
19	18	eqcomi 2745	. . 3 ⊢ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴)
20	19	sseq2i 3951	. 2 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) ↔ (𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴))
21	19	eqeq2i 2749	. 2 ⊢ (ran 𝐹 = ran (𝐹 ↾ 𝐴) ↔ ran 𝐹 = (𝐹 “ 𝐴))
22	17, 20, 21	3imtr4g 296	1 ⊢ (Fun ◡𝐹 → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∖ cdif 3886   ∪ cun 3887   ⊆ wss 3889  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   ↾ cres 5633   “ cima 5634  Fun wfun 6492

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-10 2147  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

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-nf 1786  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6500

This theorem is referenced by: (None)