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Theorem 2ndinl 9966
Description: The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.)
Assertion
Ref Expression
2ndinl (𝑋𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)

Proof of Theorem 2ndinl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-inl 9940 . . . 4 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2 opeq2 4879 . . . 4 (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
3 elex 3499 . . . 4 (𝑋𝑉𝑋 ∈ V)
4 opex 5475 . . . . 5 ⟨∅, 𝑋⟩ ∈ V
54a1i 11 . . . 4 (𝑋𝑉 → ⟨∅, 𝑋⟩ ∈ V)
61, 2, 3, 5fvmptd3 7039 . . 3 (𝑋𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
76fveq2d 6911 . 2 (𝑋𝑉 → (2nd ‘(inl‘𝑋)) = (2nd ‘⟨∅, 𝑋⟩))
8 0ex 5313 . . 3 ∅ ∈ V
9 op2ndg 8026 . . 3 ((∅ ∈ V ∧ 𝑋𝑉) → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
108, 9mpan 690 . 2 (𝑋𝑉 → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
117, 10eqtrd 2775 1 (𝑋𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  cop 4637  cfv 6563  2nd c2nd 8012  inlcinl 9937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-2nd 8014  df-inl 9940
This theorem is referenced by:  updjudhcoinlf  9970
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