Theorem rnghmresfn 20641

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 20465	. . 3 ⊢ RngHom Fn (Rng × Rng)
2		rnghmresfn.b	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
3		inss2 4259	. . . . 5 ⊢ (𝑈 ∩ Rng) ⊆ Rng
4	2, 3	eqsstrdi 4063	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ Rng)
5		xpss12 5715	. . . 4 ⊢ ((𝐵 ⊆ Rng ∧ 𝐵 ⊆ Rng) → (𝐵 × 𝐵) ⊆ (Rng × Rng))
6	4, 4, 5	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ (Rng × Rng))
7		fnssres 6703	. . 3 ⊢ (( RngHom Fn (Rng × Rng) ∧ (𝐵 × 𝐵) ⊆ (Rng × Rng)) → ( RngHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
8	1, 6, 7	sylancr 586	. 2 ⊢ (𝜑 → ( RngHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
9		rnghmresfn.h	. . 3 ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))
10	9	fneq1d 6672	. 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 3975   ⊆ wss 3976   × cxp 5698   ↾ cres 5702   Fn wfn 6568  Rngcrng 20179   RngHom crnghm 20460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-rnghm 20462

This theorem is referenced by:  rngcbas  20643  rngchomfval  20644  rngchomfeqhom  20647  rngccofval  20648  dfrngc2  20650  rnghmsubcsetc  20655  rngcid  20657  funcrngcsetc  20662