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Theorem 2ndinr 9970
Description: The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.)
Assertion
Ref Expression
2ndinr (𝑋𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)

Proof of Theorem 2ndinr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-inr 9943 . . . 4 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2 opeq2 4874 . . . 4 (𝑥 = 𝑋 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
3 elex 3501 . . . 4 (𝑋𝑉𝑋 ∈ V)
4 opex 5469 . . . . 5 ⟨1o, 𝑋⟩ ∈ V
54a1i 11 . . . 4 (𝑋𝑉 → ⟨1o, 𝑋⟩ ∈ V)
61, 2, 3, 5fvmptd3 7039 . . 3 (𝑋𝑉 → (inr‘𝑋) = ⟨1o, 𝑋⟩)
76fveq2d 6910 . 2 (𝑋𝑉 → (2nd ‘(inr‘𝑋)) = (2nd ‘⟨1o, 𝑋⟩))
8 1oex 8516 . . 3 1o ∈ V
9 op2ndg 8027 . . 3 ((1o ∈ V ∧ 𝑋𝑉) → (2nd ‘⟨1o, 𝑋⟩) = 𝑋)
108, 9mpan 690 . 2 (𝑋𝑉 → (2nd ‘⟨1o, 𝑋⟩) = 𝑋)
117, 10eqtrd 2777 1 (𝑋𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  cfv 6561  2nd c2nd 8013  1oc1o 8499  inrcinr 9940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-suc 6390  df-iota 6514  df-fun 6563  df-fv 6569  df-2nd 8015  df-1o 8506  df-inr 9943
This theorem is referenced by:  updjudhcoinrg  9973
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