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Theorem rhmresel 20542
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 480 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
32oveqd 7421 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐻𝑌) = (𝑋( RingHom ↾ (𝐵 × 𝐵))𝑌))
4 ovres 7569 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋( RingHom ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RingHom 𝑌))
54adantl 481 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋( RingHom ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RingHom 𝑌))
63, 5eqtrd 2766 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))
76eleq2d 2813 . 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 395  w3a 1084   = wceq 1533  wcel 2098   × cxp 5667  cres 5671  (class class class)co 7404   RingHom crh 20368
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-res 5681  df-iota 6488  df-fv 6544  df-ov 7407
This theorem is referenced by:  rhmsubcsetclem2  20554  rhmsubcrngclem2  20560
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