Theorem rnghmresfn 46763

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 46622	. . 3 ⊢ RngHomo Fn (Rng × Rng)
2		rnghmresfn.b	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
3		inss2 4228	. . . . 5 ⊢ (𝑈 ∩ Rng) ⊆ Rng
4	2, 3	eqsstrdi 4035	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ Rng)
5		xpss12 5690	. . . 4 ⊢ ((𝐵 ⊆ Rng ∧ 𝐵 ⊆ Rng) → (𝐵 × 𝐵) ⊆ (Rng × Rng))
6	4, 4, 5	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ (Rng × Rng))
7		fnssres 6670	. . 3 ⊢ (( RngHomo Fn (Rng × Rng) ∧ (𝐵 × 𝐵) ⊆ (Rng × Rng)) → ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
8	1, 6, 7	sylancr 588	. 2 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
9		rnghmresfn.h	. . 3 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
10	9	fneq1d 6639	. 2 ⊢ (𝜑 → (𝐻 Fn (𝐵 × 𝐵) ↔ ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)))
11	8, 10	mpbird 257	1 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))

Syntax hints:   → wi 4   = wceq 1542   ∩ cin 3946   ⊆ wss 3947   × cxp 5673   ↾ cres 5677   Fn wfn 6535  Rngcrng 46583   RngHomo crngh 46617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7970  df-2nd 7971  df-rnghomo 46619

This theorem is referenced by:  rngcbas  46765  rngchomfval  46766  rngchomfeqhom  46769  rngccofval  46770  dfrngc2  46772  rnghmsubcsetc  46777  rngcid  46779  funcrngcsetc  46798