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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 (𝜑𝐹 Fn 𝐴)
rescnvimafod.e (𝜑𝐸 = (ran 𝐹𝐵))
rescnvimafod.d (𝜑𝐷 = (𝐹𝐵))
Assertion
Ref Expression
rescnvimafod (𝜑 → (𝐹𝐷):𝐷onto𝐸)

Proof of Theorem rescnvimafod
StepHypRef Expression
1 rescnvimafod.f . . 3 (𝜑𝐹 Fn 𝐴)
2 cnvimass 6100 . . . . 5 (𝐹𝐵) ⊆ dom 𝐹
32a1i 11 . . . 4 (𝜑 → (𝐹𝐵) ⊆ dom 𝐹)
4 rescnvimafod.d . . . 4 (𝜑𝐷 = (𝐹𝐵))
51fndmd 6673 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
65eqcomd 2743 . . . 4 (𝜑𝐴 = dom 𝐹)
73, 4, 63sstr4d 4039 . . 3 (𝜑𝐷𝐴)
81, 7fnssresd 6692 . 2 (𝜑 → (𝐹𝐷) Fn 𝐷)
9 df-ima 5698 . . . 4 (𝐹𝐷) = ran (𝐹𝐷)
104imaeq2d 6078 . . . . 5 (𝜑 → (𝐹𝐷) = (𝐹 “ (𝐹𝐵)))
11 fnfun 6668 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
12 funimacnv 6647 . . . . . 6 (Fun 𝐹 → (𝐹 “ (𝐹𝐵)) = (𝐵 ∩ ran 𝐹))
131, 11, 123syl 18 . . . . 5 (𝜑 → (𝐹 “ (𝐹𝐵)) = (𝐵 ∩ ran 𝐹))
14 incom 4209 . . . . . 6 (𝐵 ∩ ran 𝐹) = (ran 𝐹𝐵)
1514a1i 11 . . . . 5 (𝜑 → (𝐵 ∩ ran 𝐹) = (ran 𝐹𝐵))
1610, 13, 153eqtrd 2781 . . . 4 (𝜑 → (𝐹𝐷) = (ran 𝐹𝐵))
179, 16eqtr3id 2791 . . 3 (𝜑 → ran (𝐹𝐷) = (ran 𝐹𝐵))
18 rescnvimafod.e . . 3 (𝜑𝐸 = (ran 𝐹𝐵))
1917, 18eqtr4d 2780 . 2 (𝜑 → ran (𝐹𝐷) = 𝐸)
20 df-fo 6567 . 2 ((𝐹𝐷):𝐷onto𝐸 ↔ ((𝐹𝐷) Fn 𝐷 ∧ ran (𝐹𝐷) = 𝐸))
218, 19, 20sylanbrc 583 1 (𝜑 → (𝐹𝐷):𝐷onto𝐸)
Colors of variables: wff setvar class
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  ontowfo 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
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