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Theorem rhmresel 45241
Description: An element of the unital ring homomorphisms restricted to a subset of unital rings is a unital ring homomorphism. (Contributed by AV, 10-Mar-2020.)
Hypothesis
Ref Expression
rhmresel.h (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
Assertion
Ref Expression
rhmresel ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RingHom 𝑌))

Proof of Theorem rhmresel
StepHypRef Expression
1 rhmresel.h . . . . . 6 (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
21adantr 484 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
32oveqd 7230 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐻𝑌) = (𝑋( RingHom ↾ (𝐵 × 𝐵))𝑌))
4 ovres 7374 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋( RingHom ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RingHom 𝑌))
54adantl 485 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋( RingHom ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RingHom 𝑌))
63, 5eqtrd 2777 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))
76eleq2d 2823 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹 ∈ (𝑋 RingHom 𝑌)))
87biimp3a 1471 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RingHom 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110   × cxp 5549  cres 5553  (class class class)co 7213   RingHom crh 19732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-xp 5557  df-res 5563  df-iota 6338  df-fv 6388  df-ov 7216
This theorem is referenced by:  rhmsubcsetclem2  45253  rhmsubcrngclem2  45259
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