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Theorem 2ndinl 9967
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 9941 . . . 4 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2 opeq2 4879 . . . 4 (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
3 elex 3481 . . . 4 (𝑋𝑉𝑋 ∈ V)
4 opex 5469 . . . . 5 ⟨∅, 𝑋⟩ ∈ V
54a1i 11 . . . 4 (𝑋𝑉 → ⟨∅, 𝑋⟩ ∈ V)
61, 2, 3, 5fvmptd3 7031 . . 3 (𝑋𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
76fveq2d 6904 . 2 (𝑋𝑉 → (2nd ‘(inl‘𝑋)) = (2nd ‘⟨∅, 𝑋⟩))
8 0ex 5311 . . 3 ∅ ∈ V
9 op2ndg 8015 . . 3 ((∅ ∈ V ∧ 𝑋𝑉) → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
108, 9mpan 688 . 2 (𝑋𝑉 → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
117, 10eqtrd 2765 1 (𝑋𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3461  c0 4324  cop 4638  cfv 6553  2nd c2nd 8001  inlcinl 9938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-2nd 8003  df-inl 9941
This theorem is referenced by:  updjudhcoinlf  9971
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