Theorem f1imadifssran 6610

Description: Condition for the range of a one-to-one function to be the range of one its restrictions. Variant of imadifssran 6132. (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 6049	. . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		imadif 6608	. . . . . . 7 ⊢ (Fun ◡𝐹 → (𝐹 “ (dom 𝐹 ∖ 𝐴)) = ((𝐹 “ dom 𝐹) ∖ (𝐹 “ 𝐴)))
3	2	sseq1d 3986	. . . . . 6 ⊢ (Fun ◡𝐹 → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴) ↔ ((𝐹 “ dom 𝐹) ∖ (𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐴)))
4		ssundif 4459	. . . . . . 7 ⊢ ((𝐹 “ dom 𝐹) ⊆ ((𝐹 “ 𝐴) ∪ (𝐹 “ 𝐴)) ↔ ((𝐹 “ dom 𝐹) ∖ (𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐴))
5		unidm 4128	. . . . . . . . 9 ⊢ ((𝐹 “ 𝐴) ∪ (𝐹 “ 𝐴)) = (𝐹 “ 𝐴)
6	5	sseq2i 3984	. . . . . . . 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 6050	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
9	8, 1	sseqtrri 4004	. . . . . . . . . 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 3972	. . . . . . . 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 2779	. . 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 5659	. . . 4 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
19	18	eqcomi 2739	. . 3 ⊢ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴)
20	19	sseq2i 3984	. 2 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) ↔ (𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴))
21	19	eqeq2i 2743	. 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 1540   ∖ cdif 3919   ∪ cun 3920   ⊆ wss 3922  ^◡ccnv 5645  dom cdm 5646  ran crn 5647   ↾ cres 5648   “ cima 5649  Fun wfun 6513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-br 5116  df-opab 5178  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-fun 6521

This theorem is referenced by: (None)