Theorem rnghmresfn 41224

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 41151	. . 3 ⊢ RngHomo Fn (Rng × Rng)
2		rnghmresfn.b	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
3		inss2 3817	. . . . 5 ⊢ (𝑈 ∩ Rng) ⊆ Rng
4	2, 3	syl6eqss 3639	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ Rng)
5		xpss12 5191	. . . 4 ⊢ ((𝐵 ⊆ Rng ∧ 𝐵 ⊆ Rng) → (𝐵 × 𝐵) ⊆ (Rng × Rng))
6	4, 4, 5	syl2anc 692	. . 3 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ (Rng × Rng))
7		fnssres 5964	. . 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 5941	. 2 ⊢ (𝜑 → (𝐻 Fn (𝐵 × 𝐵) ↔ ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)))
11	8, 10	mpbird 247	1 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))

Syntax hints:   → wi 4   = wceq 1480   ∩ cin 3559   ⊆ wss 3560   × cxp 5077   ↾ cres 5081   Fn wfn 5845  Rngcrng 41135   RngHomo crngh 41146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-rnghomo 41148

This theorem is referenced by:  rngcbas  41226  rngchomfval  41227  rngchomfeqhom  41230  rngccofval  41231  dfrngc2  41233  rnghmsubcsetc  41238  rngcid  41240  funcrngcsetc  41259