Theorem f1imadifssran 6602

Description: Condition for the range of a one-to-one function to be the range of one its restrictions. Variant of imadifssran 6124. (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 6041	. . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		imadif 6600	. . . . . . 7 ⊢ (Fun ◡𝐹 → (𝐹 “ (dom 𝐹 ∖ 𝐴)) = ((𝐹 “ dom 𝐹) ∖ (𝐹 “ 𝐴)))
3	2	sseq1d 3978	. . . . . 6 ⊢ (Fun ◡𝐹 → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴) ↔ ((𝐹 “ dom 𝐹) ∖ (𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐴)))
4		ssundif 4451	. . . . . . 7 ⊢ ((𝐹 “ dom 𝐹) ⊆ ((𝐹 “ 𝐴) ∪ (𝐹 “ 𝐴)) ↔ ((𝐹 “ dom 𝐹) ∖ (𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐴))
5		unidm 4120	. . . . . . . . 9 ⊢ ((𝐹 “ 𝐴) ∪ (𝐹 “ 𝐴)) = (𝐹 “ 𝐴)
6	5	sseq2i 3976	. . . . . . . 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 6042	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
9	8, 1	sseqtrri 3996	. . . . . . . . . 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 3964	. . . . . . . 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 2778	. . 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 5651	. . . 4 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
19	18	eqcomi 2738	. . 3 ⊢ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴)
20	19	sseq2i 3976	. 2 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) ↔ (𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴))
21	19	eqeq2i 2742	. 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 3911   ∪ cun 3912   ⊆ wss 3914  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   ↾ cres 5640   “ cima 5641  Fun wfun 6505

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-fun 6513

This theorem is referenced by: (None)