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Theorem 2ndinl 6846
 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 6819 . . . . 5 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
21a1i 9 . . . 4 (𝑋𝑉 → inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩))
3 opeq2 3645 . . . . 5 (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
43adantl 272 . . . 4 ((𝑋𝑉𝑥 = 𝑋) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
5 elex 2644 . . . 4 (𝑋𝑉𝑋 ∈ V)
6 0ex 3987 . . . . 5 ∅ ∈ V
7 opexg 4079 . . . . 5 ((∅ ∈ V ∧ 𝑋𝑉) → ⟨∅, 𝑋⟩ ∈ V)
86, 7mpan 416 . . . 4 (𝑋𝑉 → ⟨∅, 𝑋⟩ ∈ V)
92, 4, 5, 8fvmptd 5420 . . 3 (𝑋𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
109fveq2d 5344 . 2 (𝑋𝑉 → (2nd ‘(inl‘𝑋)) = (2nd ‘⟨∅, 𝑋⟩))
11 op2ndg 5960 . . 3 ((∅ ∈ V ∧ 𝑋𝑉) → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
126, 11mpan 416 . 2 (𝑋𝑉 → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
1310, 12eqtrd 2127 1 (𝑋𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1296   ∈ wcel 1445  Vcvv 2633  ∅c0 3302  ⟨cop 3469   ↦ cmpt 3921  ‘cfv 5049  2nd c2nd 5948  inlcinl 6817 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-iota 5014  df-fun 5051  df-fv 5057  df-2nd 5950  df-inl 6819 This theorem is referenced by:  updjudhcoinlf  6851
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