Theorem rnghmresfn 41755

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 = ( RngHomo ↾ (𝐵 × 𝐵)))

Assertion

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

Proof of Theorem rnghmresfn

Step	Hyp	Ref	Expression
1		rnghmfn 41680	. . 3 ⊢ RngHomo Fn (Rng × Rng)
2		rnghmresfn.b	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
3		inss2 3796	. . . . 5 ⊢ (𝑈 ∩ Rng) ⊆ Rng
4	2, 3	syl6eqss 3618	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ Rng)
5		xpss12 5148	. . . 4 ⊢ ((𝐵 ⊆ Rng ∧ 𝐵 ⊆ Rng) → (𝐵 × 𝐵) ⊆ (Rng × Rng))
6	4, 4, 5	syl2anc 691	. . 3 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ (Rng × Rng))
7		fnssres 5918	. . 3 ⊢ (( RngHomo Fn (Rng × Rng) ∧ (𝐵 × 𝐵) ⊆ (Rng × Rng)) → ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
8	1, 6, 7	sylancr 694	. 2 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
9		rnghmresfn.h	. . 3 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
10	9	fneq1d 5895	. 2 ⊢ (𝜑 → (𝐻 Fn (𝐵 × 𝐵) ↔ ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)))
11	8, 10	mpbird 246	1 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))

Syntax hints:   → wi 4   = wceq 1475   ∩ cin 3539   ⊆ wss 3540   × cxp 5036   ↾ cres 5040   Fn wfn 5799  Rngcrng 41664   RngHomo crngh 41675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-rnghomo 41677

This theorem is referenced by:  rngcbas  41757  rngchomfval  41758  rngchomfeqhom  41761  rngccofval  41762  dfrngc2  41764  rnghmsubcsetc  41769  rngcid  41771  funcrngcsetc  41790