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Theorem rnghmresel 42993
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 6941 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐻𝑌) = (𝑋( RngHomo ↾ (𝐵 × 𝐵))𝑌))
4 ovres 7079 . . . . 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 2814 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐻𝑌) = (𝑋 RngHomo 𝑌))
76eleq2d 2845 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹 ∈ (𝑋 RngHomo 𝑌)))
87biimp3a 1542 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RngHomo 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107   × cxp 5355  cres 5359  (class class class)co 6924   RngHomo crngh 42914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-xp 5363  df-res 5369  df-iota 6101  df-fv 6145  df-ov 6927
This theorem is referenced by:  rnghmsubcsetclem2  43005
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