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Theorem rnghmresel 42804
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 (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
Assertion
Ref Expression
rnghmresel ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RngHomo 𝑌))

Proof of Theorem rnghmresel
StepHypRef Expression
1 rnghmresel.h . . . . . 6 (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
21adantr 474 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
32oveqd 6922 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐻𝑌) = (𝑋( RngHomo ↾ (𝐵 × 𝐵))𝑌))
4 ovres 7060 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋( RngHomo ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RngHomo 𝑌))
54adantl 475 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋( RngHomo ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RngHomo 𝑌))
63, 5eqtrd 2861 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐻𝑌) = (𝑋 RngHomo 𝑌))
76eleq2d 2892 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹 ∈ (𝑋 RngHomo 𝑌)))
87biimp3a 1597 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RngHomo 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1111   = wceq 1656  wcel 2164   × cxp 5340  cres 5344  (class class class)co 6905   RngHomo crngh 42725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-xp 5348  df-res 5354  df-iota 6086  df-fv 6131  df-ov 6908
This theorem is referenced by:  rnghmsubcsetclem2  42816
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