Theorem mat1rhmval 22800

Description: The value of the ring homomorphism 𝐹. (Contributed by AV, 22-Dec-2019.)

Hypotheses

Ref	Expression
mat1rhmval.k	⊢ 𝐾 = (Base‘𝑅)
mat1rhmval.a	⊢ 𝐴 = ({𝐸} Mat 𝑅)
mat1rhmval.b	⊢ 𝐵 = (Base‘𝐴)
mat1rhmval.o	⊢ 𝑂 = ⟨𝐸, 𝐸⟩
mat1rhmval.f	⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {⟨𝑂, 𝑥⟩})

Assertion

Ref	Expression
mat1rhmval	⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = {⟨𝑂, 𝑋⟩})

Distinct variable groups:   𝑥,𝐾   𝑥,𝑂   𝑥,𝐸   𝑥,𝑅   𝑥,𝑉   𝑥,𝑋

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem mat1rhmval

Step	Hyp	Ref	Expression
1		mat1rhmval.f	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {⟨𝑂, 𝑥⟩})
2		opeq2 4840	. . 3 ⊢ (𝑥 = 𝑋 → ⟨𝑂, 𝑥⟩ = ⟨𝑂, 𝑋⟩)
3	2	sneqd 4592	. 2 ⊢ (𝑥 = 𝑋 → {⟨𝑂, 𝑥⟩} = {⟨𝑂, 𝑋⟩})
4		simp3 1156	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾)
5		snex 5396	. . 3 ⊢ {⟨𝑂, 𝑋⟩} ∈ V
6	5	a1i 11	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → {⟨𝑂, 𝑋⟩} ∈ V)
7	1, 3, 4, 6	fvmptd3 7018	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = {⟨𝑂, 𝑋⟩})

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  Vcvv 3450  {csn 4584  ⟨cop 4834   ↦ cmpt 5185  ‘cfv 6539  (class class class)co 7421  Basecbs 17393  Ringcrg 20465   Mat cmat 22728

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547

This theorem is used by:  mat1rhmelval  22801  mat1rhmcl  22802  mat1mhm  22805