Theorem rnghmresel 20872

Description: An element of the non-unital ring homomorphisms restricted to a subset of non-unital rings is a non-unital ring homomorphisms. (Contributed by AV, 9-Mar-2020.)

Hypothesis

Ref	Expression
rnghmresel.h	⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))

Assertion

Ref	Expression
rnghmresel	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RngHom 𝑌))

Proof of Theorem rnghmresel

Step	Hyp	Ref	Expression
1		rnghmresel.h	. . . . . 6 ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))
2	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))
3	2	oveqd 7437	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐻𝑌) = (𝑋( RngHom ↾ (𝐵 × 𝐵))𝑌))
4		ovres 7586	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋( RngHom ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RngHom 𝑌))
5	4	adantl 487	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋( RngHom ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RngHom 𝑌))
6	3, 5	eqtrd 2795	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐻𝑌) = (𝑋 RngHom 𝑌))
7	6	eleq2d 2846	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹 ∈ (𝑋 RngHom 𝑌)))
8	7	biimp3a 1498	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RngHom 𝑌))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   × cxp 5648   ↾ cres 5652  (class class class)co 7420   RngHom crnghm 20664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-res 5662  df-iota 6494  df-fv 6546  df-ov 7423

This theorem is used by:  rnghmsubcsetclem2  20884