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Theorem rhmresel 20589
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 479 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
32oveqd 7443 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐻𝑌) = (𝑋( RingHom ↾ (𝐵 × 𝐵))𝑌))
4 ovres 7593 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋( RingHom ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RingHom 𝑌))
54adantl 480 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋( RingHom ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RingHom 𝑌))
63, 5eqtrd 2768 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))
76eleq2d 2815 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹 ∈ (𝑋 RingHom 𝑌)))
87biimp3a 1465 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RingHom 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098   × cxp 5680  cres 5684  (class class class)co 7426   RingHom crh 20415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5688  df-res 5694  df-iota 6505  df-fv 6561  df-ov 7429
This theorem is referenced by:  rhmsubcsetclem2  20601  rhmsubcrngclem2  20607
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