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Theorem rnghmresel 45076
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 484 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
32oveqd 7187 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐻𝑌) = (𝑋( RngHomo ↾ (𝐵 × 𝐵))𝑌))
4 ovres 7330 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋( RngHomo ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RngHomo 𝑌))
54adantl 485 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋( RngHomo ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RngHomo 𝑌))
63, 5eqtrd 2773 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐻𝑌) = (𝑋 RngHomo 𝑌))
76eleq2d 2818 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹 ∈ (𝑋 RngHomo 𝑌)))
87biimp3a 1470 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RngHomo 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114   × cxp 5523  cres 5527  (class class class)co 7170   RngHomo crngh 44997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-xp 5531  df-res 5537  df-iota 6297  df-fv 6347  df-ov 7173
This theorem is referenced by:  rnghmsubcsetclem2  45088
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