Theorem rescnvimafod 7093

Description: The restriction of a function to a preimage of a class is a function onto the intersection of this class and the range of the function. (Contributed by AV, 13-Sep-2024.) (Revised by AV, 29-Sep-2024.)

Hypotheses

Ref	Expression
rescnvimafod.f	⊢ (𝜑 → 𝐹 Fn 𝐴)
rescnvimafod.e	⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐵))
rescnvimafod.d	⊢ (𝜑 → 𝐷 = (◡𝐹 “ 𝐵))

Assertion

Ref	Expression
rescnvimafod	⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→𝐸)

Proof of Theorem rescnvimafod

Step	Hyp	Ref	Expression
1		rescnvimafod.f	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		cnvimass 6100	. . . . 5 ⊢ (◡𝐹 “ 𝐵) ⊆ dom 𝐹
3	2	a1i 11	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝐵) ⊆ dom 𝐹)
4		rescnvimafod.d	. . . 4 ⊢ (𝜑 → 𝐷 = (◡𝐹 “ 𝐵))
5	1	fndmd 6673	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
6	5	eqcomd 2743	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
7	3, 4, 6	3sstr4d 4039	. . 3 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
8	1, 7	fnssresd 6692	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐷) Fn 𝐷)
9		df-ima 5698	. . . 4 ⊢ (𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷)
10	4	imaeq2d 6078	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐷) = (𝐹 “ (◡𝐹 “ 𝐵)))
11		fnfun 6668	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
12		funimacnv 6647	. . . . . 6 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐵)) = (𝐵 ∩ ran 𝐹))
13	1, 11, 12	3syl 18	. . . . 5 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝐵)) = (𝐵 ∩ ran 𝐹))
14		incom 4209	. . . . . 6 ⊢ (𝐵 ∩ ran 𝐹) = (ran 𝐹 ∩ 𝐵)
15	14	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐵 ∩ ran 𝐹) = (ran 𝐹 ∩ 𝐵))
16	10, 13, 15	3eqtrd 2781	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝐷) = (ran 𝐹 ∩ 𝐵))
17	9, 16	eqtr3id 2791	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = (ran 𝐹 ∩ 𝐵))
18		rescnvimafod.e	. . 3 ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐵))
19	17, 18	eqtr4d 2780	. 2 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = 𝐸)
20		df-fo 6567	. 2 ⊢ ((𝐹 ↾ 𝐷):𝐷–onto→𝐸 ↔ ((𝐹 ↾ 𝐷) Fn 𝐷 ∧ ran (𝐹 ↾ 𝐷) = 𝐸))
21	8, 19, 20	sylanbrc 583	1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→𝐸)

Syntax hints:   → wi 4   = wceq 1540   ∩ cin 3950   ⊆ wss 3951  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   ↾ cres 5687   “ cima 5688  Fun wfun 6555   Fn wfn 6556  –onto→wfo 6559

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 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-fo 6567

This theorem is referenced by:  fresfo  47060  fcoreslem3  47077  3f1oss1  47087