Theorem f1imadifssran 6578

Description: Condition for the range of a one-to-one function to be the range of one its restrictions. Variant of imadifssran 6109. (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 6029	. . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		imadif 6576	. . . . . . 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 4422	. . . . . . 7 ⊢ ((𝐹 “ dom 𝐹) ⊆ ((𝐹 “ 𝐴) ∪ (𝐹 “ 𝐴)) ↔ ((𝐹 “ dom 𝐹) ∖ (𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐴))
5		unidm 4094	. . . . . . . . 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 6030	. . . . . . . . . . 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 218	. . . . . . 7 ⊢ ((𝐹 “ dom 𝐹) ⊆ ((𝐹 “ 𝐴) ∪ (𝐹 “ 𝐴)) → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
13	4, 12	sylbir 236	. . . . . 6 ⊢ (((𝐹 “ dom 𝐹) ∖ (𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐴) → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
14	3, 13	biimtrdi 254	. . . . 5 ⊢ (Fun ◡𝐹 → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴) → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴)))
15	14	imp 407	. . . 4 ⊢ ((Fun ◡𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴)) → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
16	1, 15	eqtr3id 2789	. . 3 ⊢ ((Fun ◡𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴)) → ran 𝐹 = (𝐹 “ 𝐴))
17	16	ex 413	. 2 ⊢ (Fun ◡𝐹 → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴) → ran 𝐹 = (𝐹 “ 𝐴)))
18		df-ima 5638	. . . 4 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
19	18	eqcomi 2749	. . 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 2753	. 2 ⊢ (ran 𝐹 = ran (𝐹 ↾ 𝐴) ↔ ran 𝐹 = (𝐹 “ 𝐴))
22	17, 20, 21	3imtr4g 297	1 ⊢ (Fun ◡𝐹 → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∖ cdif 3887   ∪ cun 3888   ⊆ wss 3890  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   ↾ cres 5627   “ cima 5628  Fun wfun 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-fun 6494

This theorem is referenced by: (None)