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Theorem rhmresel 42535
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 466 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
32oveqd 6813 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐻𝑌) = (𝑋( RingHom ↾ (𝐵 × 𝐵))𝑌))
4 ovres 6951 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋( RingHom ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RingHom 𝑌))
54adantl 467 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋( RingHom ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RingHom 𝑌))
63, 5eqtrd 2805 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))
76eleq2d 2836 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹 ∈ (𝑋 RingHom 𝑌)))
87biimp3a 1580 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RingHom 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145   × cxp 5248  cres 5252  (class class class)co 6796   RingHom crh 18922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-xp 5256  df-res 5262  df-iota 5993  df-fv 6038  df-ov 6799
This theorem is referenced by:  rhmsubcsetclem2  42547  rhmsubcrngclem2  42553
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