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Theorem 2ndinl 7050
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 7022 . . . . 5 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
21a1i 9 . . . 4 (𝑋𝑉 → inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩))
3 opeq2 3764 . . . . 5 (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
43adantl 275 . . . 4 ((𝑋𝑉𝑥 = 𝑋) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
5 elex 2741 . . . 4 (𝑋𝑉𝑋 ∈ V)
6 0ex 4114 . . . . 5 ∅ ∈ V
7 opexg 4211 . . . . 5 ((∅ ∈ V ∧ 𝑋𝑉) → ⟨∅, 𝑋⟩ ∈ V)
86, 7mpan 422 . . . 4 (𝑋𝑉 → ⟨∅, 𝑋⟩ ∈ V)
92, 4, 5, 8fvmptd 5575 . . 3 (𝑋𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
109fveq2d 5498 . 2 (𝑋𝑉 → (2nd ‘(inl‘𝑋)) = (2nd ‘⟨∅, 𝑋⟩))
11 op2ndg 6128 . . 3 ((∅ ∈ V ∧ 𝑋𝑉) → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
126, 11mpan 422 . 2 (𝑋𝑉 → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
1310, 12eqtrd 2203 1 (𝑋𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wcel 2141  Vcvv 2730  c0 3414  cop 3584  cmpt 4048  cfv 5196  2nd c2nd 6116  inlcinl 7020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fv 5204  df-2nd 6118  df-inl 7022
This theorem is referenced by:  updjudhcoinlf  7055
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