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Theorem 2ndinl 7105
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 7077 . . . . 5 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
21a1i 9 . . . 4 (𝑋𝑉 → inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩))
3 opeq2 3794 . . . . 5 (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
43adantl 277 . . . 4 ((𝑋𝑉𝑥 = 𝑋) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
5 elex 2763 . . . 4 (𝑋𝑉𝑋 ∈ V)
6 0ex 4145 . . . . 5 ∅ ∈ V
7 opexg 4246 . . . . 5 ((∅ ∈ V ∧ 𝑋𝑉) → ⟨∅, 𝑋⟩ ∈ V)
86, 7mpan 424 . . . 4 (𝑋𝑉 → ⟨∅, 𝑋⟩ ∈ V)
92, 4, 5, 8fvmptd 5618 . . 3 (𝑋𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
109fveq2d 5538 . 2 (𝑋𝑉 → (2nd ‘(inl‘𝑋)) = (2nd ‘⟨∅, 𝑋⟩))
11 op2ndg 6177 . . 3 ((∅ ∈ V ∧ 𝑋𝑉) → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
126, 11mpan 424 . 2 (𝑋𝑉 → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
1310, 12eqtrd 2222 1 (𝑋𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  Vcvv 2752  c0 3437  cop 3610  cmpt 4079  cfv 5235  2nd c2nd 6165  inlcinl 7075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fv 5243  df-2nd 6167  df-inl 7077
This theorem is referenced by:  updjudhcoinlf  7110
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