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Theorem rnghmresfn 46851
Description: The class of non-unital ring homomorphisms restricted to subsets of non-unital rings is a function. (Contributed by AV, 4-Mar-2020.)
Hypotheses
Ref Expression
rnghmresfn.b (𝜑𝐵 = (𝑈 ∩ Rng))
rnghmresfn.h (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
Assertion
Ref Expression
rnghmresfn (𝜑𝐻 Fn (𝐵 × 𝐵))

Proof of Theorem rnghmresfn
StepHypRef Expression
1 rnghmfn 46678 . . 3 RngHomo Fn (Rng × Rng)
2 rnghmresfn.b . . . . 5 (𝜑𝐵 = (𝑈 ∩ Rng))
3 inss2 4229 . . . . 5 (𝑈 ∩ Rng) ⊆ Rng
42, 3eqsstrdi 4036 . . . 4 (𝜑𝐵 ⊆ Rng)
5 xpss12 5691 . . . 4 ((𝐵 ⊆ Rng ∧ 𝐵 ⊆ Rng) → (𝐵 × 𝐵) ⊆ (Rng × Rng))
64, 4, 5syl2anc 584 . . 3 (𝜑 → (𝐵 × 𝐵) ⊆ (Rng × Rng))
7 fnssres 6673 . . 3 (( RngHomo Fn (Rng × Rng) ∧ (𝐵 × 𝐵) ⊆ (Rng × Rng)) → ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
81, 6, 7sylancr 587 . 2 (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
9 rnghmresfn.h . . 3 (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
109fneq1d 6642 . 2 (𝜑 → (𝐻 Fn (𝐵 × 𝐵) ↔ ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)))
118, 10mpbird 256 1 (𝜑𝐻 Fn (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  cin 3947  wss 3948   × cxp 5674  cres 5678   Fn wfn 6538  Rngcrng 46638   RngHomo crngh 46673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-rnghomo 46675
This theorem is referenced by:  rngcbas  46853  rngchomfval  46854  rngchomfeqhom  46857  rngccofval  46858  dfrngc2  46860  rnghmsubcsetc  46865  rngcid  46867  funcrngcsetc  46886
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