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Theorem eqresfnbd 42252
Description: Property of being the restriction of a function. Note that this is closer to funssres 6612 than fnssres 6692. (Contributed by SN, 11-Mar-2025.)
Hypotheses
Ref Expression
eqresfnbd.g (𝜑𝐹 Fn 𝐵)
eqresfnbd.1 (𝜑𝐴𝐵)
Assertion
Ref Expression
eqresfnbd (𝜑 → (𝑅 = (𝐹𝐴) ↔ (𝑅 Fn 𝐴𝑅𝐹)))

Proof of Theorem eqresfnbd
StepHypRef Expression
1 eqresfnbd.g . . . . 5 (𝜑𝐹 Fn 𝐵)
2 eqresfnbd.1 . . . . 5 (𝜑𝐴𝐵)
31, 2fnssresd 6693 . . . 4 (𝜑 → (𝐹𝐴) Fn 𝐴)
4 resss 6022 . . . 4 (𝐹𝐴) ⊆ 𝐹
53, 4jctir 520 . . 3 (𝜑 → ((𝐹𝐴) Fn 𝐴 ∧ (𝐹𝐴) ⊆ 𝐹))
6 fneq1 6660 . . . 4 (𝑅 = (𝐹𝐴) → (𝑅 Fn 𝐴 ↔ (𝐹𝐴) Fn 𝐴))
7 sseq1 4021 . . . 4 (𝑅 = (𝐹𝐴) → (𝑅𝐹 ↔ (𝐹𝐴) ⊆ 𝐹))
86, 7anbi12d 632 . . 3 (𝑅 = (𝐹𝐴) → ((𝑅 Fn 𝐴𝑅𝐹) ↔ ((𝐹𝐴) Fn 𝐴 ∧ (𝐹𝐴) ⊆ 𝐹)))
95, 8syl5ibrcom 247 . 2 (𝜑 → (𝑅 = (𝐹𝐴) → (𝑅 Fn 𝐴𝑅𝐹)))
101fnfund 6670 . . . . 5 (𝜑 → Fun 𝐹)
1110adantr 480 . . . 4 ((𝜑𝑅 Fn 𝐴) → Fun 𝐹)
12 funssres 6612 . . . . . 6 ((Fun 𝐹𝑅𝐹) → (𝐹 ↾ dom 𝑅) = 𝑅)
1312eqcomd 2741 . . . . 5 ((Fun 𝐹𝑅𝐹) → 𝑅 = (𝐹 ↾ dom 𝑅))
14 fndm 6672 . . . . . . . 8 (𝑅 Fn 𝐴 → dom 𝑅 = 𝐴)
1514adantl 481 . . . . . . 7 ((𝜑𝑅 Fn 𝐴) → dom 𝑅 = 𝐴)
1615reseq2d 6000 . . . . . 6 ((𝜑𝑅 Fn 𝐴) → (𝐹 ↾ dom 𝑅) = (𝐹𝐴))
1716eqeq2d 2746 . . . . 5 ((𝜑𝑅 Fn 𝐴) → (𝑅 = (𝐹 ↾ dom 𝑅) ↔ 𝑅 = (𝐹𝐴)))
1813, 17imbitrid 244 . . . 4 ((𝜑𝑅 Fn 𝐴) → ((Fun 𝐹𝑅𝐹) → 𝑅 = (𝐹𝐴)))
1911, 18mpand 695 . . 3 ((𝜑𝑅 Fn 𝐴) → (𝑅𝐹𝑅 = (𝐹𝐴)))
2019expimpd 453 . 2 (𝜑 → ((𝑅 Fn 𝐴𝑅𝐹) → 𝑅 = (𝐹𝐴)))
219, 20impbid 212 1 (𝜑 → (𝑅 = (𝐹𝐴) ↔ (𝑅 Fn 𝐴𝑅𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wss 3963  dom cdm 5689  cres 5691  Fun wfun 6557   Fn wfn 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-fun 6565  df-fn 6566
This theorem is referenced by: (None)
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