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Theorem rnghmresfn 42358
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 42285 . . 3 RngHomo Fn (Rng × Rng)
2 rnghmresfn.b . . . . 5 (𝜑𝐵 = (𝑈 ∩ Rng))
3 inss2 3910 . . . . 5 (𝑈 ∩ Rng) ⊆ Rng
42, 3syl6eqss 3729 . . . 4 (𝜑𝐵 ⊆ Rng)
5 xpss12 5201 . . . 4 ((𝐵 ⊆ Rng ∧ 𝐵 ⊆ Rng) → (𝐵 × 𝐵) ⊆ (Rng × Rng))
64, 4, 5syl2anc 696 . . 3 (𝜑 → (𝐵 × 𝐵) ⊆ (Rng × Rng))
7 fnssres 6085 . . 3 (( RngHomo Fn (Rng × Rng) ∧ (𝐵 × 𝐵) ⊆ (Rng × Rng)) → ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
81, 6, 7sylancr 698 . 2 (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
9 rnghmresfn.h . . 3 (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
109fneq1d 6062 . 2 (𝜑 → (𝐻 Fn (𝐵 × 𝐵) ↔ ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)))
118, 10mpbird 247 1 (𝜑𝐻 Fn (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1564  cin 3647  wss 3648   × cxp 5184  cres 5188   Fn wfn 5964  Rngcrng 42269   RngHomo crngh 42280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-fal 1570  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-rab 2991  df-v 3274  df-sbc 3510  df-csb 3608  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-nul 3992  df-if 4163  df-sn 4254  df-pr 4256  df-op 4260  df-uni 4513  df-iun 4598  df-br 4729  df-opab 4789  df-mpt 4806  df-id 5096  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-ima 5199  df-iota 5932  df-fun 5971  df-fn 5972  df-f 5973  df-fv 5977  df-ov 6736  df-oprab 6737  df-mpt2 6738  df-1st 7253  df-2nd 7254  df-rnghomo 42282
This theorem is referenced by:  rngcbas  42360  rngchomfval  42361  rngchomfeqhom  42364  rngccofval  42365  dfrngc2  42367  rnghmsubcsetc  42372  rngcid  42374  funcrngcsetc  42393
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