Theorem f1imadifssran 6572

Description: Condition for the range of a one-to-one function to be the range of one its restrictions. Variant of imadifssran 6103. (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 6023	. . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		imadif 6570	. . . . . . 7 ⊢ (Fun ◡𝐹 → (𝐹 “ (dom 𝐹 ∖ 𝐴)) = ((𝐹 “ dom 𝐹) ∖ (𝐹 “ 𝐴)))
3	2	sseq1d 3962	. . . . . 6 ⊢ (Fun ◡𝐹 → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴) ↔ ((𝐹 “ dom 𝐹) ∖ (𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐴)))
4		ssundif 4437	. . . . . . 7 ⊢ ((𝐹 “ dom 𝐹) ⊆ ((𝐹 “ 𝐴) ∪ (𝐹 “ 𝐴)) ↔ ((𝐹 “ dom 𝐹) ∖ (𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐴))
5		unidm 4106	. . . . . . . . 9 ⊢ ((𝐹 “ 𝐴) ∪ (𝐹 “ 𝐴)) = (𝐹 “ 𝐴)
6	5	sseq2i 3960	. . . . . . . 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 6024	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
9	8, 1	sseqtrri 3980	. . . . . . . . . 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 3948	. . . . . . . 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 2782	. . 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 5632	. . . 4 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
19	18	eqcomi 2742	. . 3 ⊢ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴)
20	19	sseq2i 3960	. 2 ⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) ↔ (𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ (𝐹 “ 𝐴))
21	19	eqeq2i 2746	. 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 1541   ∖ cdif 3895   ∪ cun 3896   ⊆ wss 3898  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   ↾ cres 5621   “ cima 5622  Fun wfun 6480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-fun 6488

This theorem is referenced by: (None)