Theorem rnghmresfn 20636

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	⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
rnghmresfn.h	⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))

Assertion

Ref	Expression
rnghmresfn	⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))

Proof of Theorem rnghmresfn

Step	Hyp	Ref	Expression
1		rnghmfn 20456	. . 3 ⊢ RngHom Fn (Rng × Rng)
2		rnghmresfn.b	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
3		inss2 4246	. . . . 5 ⊢ (𝑈 ∩ Rng) ⊆ Rng
4	2, 3	eqsstrdi 4050	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ Rng)
5		xpss12 5704	. . . 4 ⊢ ((𝐵 ⊆ Rng ∧ 𝐵 ⊆ Rng) → (𝐵 × 𝐵) ⊆ (Rng × Rng))
6	4, 4, 5	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ (Rng × Rng))
7		fnssres 6692	. . 3 ⊢ (( RngHom Fn (Rng × Rng) ∧ (𝐵 × 𝐵) ⊆ (Rng × Rng)) → ( RngHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
8	1, 6, 7	sylancr 587	. 2 ⊢ (𝜑 → ( RngHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
9		rnghmresfn.h	. . 3 ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))
10	9	fneq1d 6662	. 2 ⊢ (𝜑 → (𝐻 Fn (𝐵 × 𝐵) ↔ ( RngHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)))
11	8, 10	mpbird 257	1 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))

Syntax hints:   → wi 4   = wceq 1537   ∩ cin 3962   ⊆ wss 3963   × cxp 5687   ↾ cres 5691   Fn wfn 6558  Rngcrng 20170   RngHom crnghm 20451

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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

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-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-rnghm 20453

This theorem is referenced by:  rngcbas  20638  rngchomfval  20639  rngchomfeqhom  20642  rngccofval  20643  dfrngc2  20645  rnghmsubcsetc  20650  rngcid  20652  funcrngcsetc  20657