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Theorem 1stinl 6927
Description: The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.)
Assertion
Ref Expression
1stinl (𝑋𝑉 → (1st ‘(inl‘𝑋)) = ∅)

Proof of Theorem 1stinl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-inl 6900 . . . . 5 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
21a1i 9 . . . 4 (𝑋𝑉 → inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩))
3 opeq2 3676 . . . . 5 (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
43adantl 275 . . . 4 ((𝑋𝑉𝑥 = 𝑋) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
5 elex 2671 . . . 4 (𝑋𝑉𝑋 ∈ V)
6 0ex 4025 . . . . 5 ∅ ∈ V
7 opexg 4120 . . . . 5 ((∅ ∈ V ∧ 𝑋𝑉) → ⟨∅, 𝑋⟩ ∈ V)
86, 7mpan 420 . . . 4 (𝑋𝑉 → ⟨∅, 𝑋⟩ ∈ V)
92, 4, 5, 8fvmptd 5470 . . 3 (𝑋𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
109fveq2d 5393 . 2 (𝑋𝑉 → (1st ‘(inl‘𝑋)) = (1st ‘⟨∅, 𝑋⟩))
11 op1stg 6016 . . 3 ((∅ ∈ V ∧ 𝑋𝑉) → (1st ‘⟨∅, 𝑋⟩) = ∅)
126, 11mpan 420 . 2 (𝑋𝑉 → (1st ‘⟨∅, 𝑋⟩) = ∅)
1310, 12eqtrd 2150 1 (𝑋𝑉 → (1st ‘(inl‘𝑋)) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wcel 1465  Vcvv 2660  c0 3333  cop 3500  cmpt 3959  cfv 5093  1st c1st 6004  inlcinl 6898
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fv 5101  df-1st 6006  df-inl 6900
This theorem is referenced by:  djune  6931  updjudhcoinlf  6933  subctctexmid  13123
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