Theorem rnghmresel 20728

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 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))
3	2	oveqd 7427	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐻𝑌) = (𝑋( RngHom ↾ (𝐵 × 𝐵))𝑌))
4		ovres 7576	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋( RngHom ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RngHom 𝑌))
5	4	adantl 486	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋( RngHom ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RngHom 𝑌))
6	3, 5	eqtrd 2798	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐻𝑌) = (𝑋 RngHom 𝑌))
7	6	eleq2d 2849	. 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 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   × cxp 5659   ↾ cres 5663  (class class class)co 7410   RngHom crnghm 20521

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-xp 5667  df-res 5673  df-iota 6492  df-fv 6544  df-ov 7413

This theorem is used by:  rnghmsubcsetclem2  20740