ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  1stinl GIF version

Theorem 1stinl 6763
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 6737 . . . . 5 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
21a1i 9 . . . 4 (𝑋𝑉 → inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩))
3 opeq2 3623 . . . . 5 (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
43adantl 271 . . . 4 ((𝑋𝑉𝑥 = 𝑋) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
5 elex 2630 . . . 4 (𝑋𝑉𝑋 ∈ V)
6 0ex 3966 . . . . 5 ∅ ∈ V
7 opexg 4055 . . . . 5 ((∅ ∈ V ∧ 𝑋𝑉) → ⟨∅, 𝑋⟩ ∈ V)
86, 7mpan 415 . . . 4 (𝑋𝑉 → ⟨∅, 𝑋⟩ ∈ V)
92, 4, 5, 8fvmptd 5385 . . 3 (𝑋𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
109fveq2d 5309 . 2 (𝑋𝑉 → (1st ‘(inl‘𝑋)) = (1st ‘⟨∅, 𝑋⟩))
11 op1stg 5921 . . 3 ((∅ ∈ V ∧ 𝑋𝑉) → (1st ‘⟨∅, 𝑋⟩) = ∅)
126, 11mpan 415 . 2 (𝑋𝑉 → (1st ‘⟨∅, 𝑋⟩) = ∅)
1310, 12eqtrd 2120 1 (𝑋𝑉 → (1st ‘(inl‘𝑋)) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wcel 1438  Vcvv 2619  c0 3286  cop 3449  cmpt 3899  cfv 5015  1st c1st 5909  inlcinl 6735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-1st 5911  df-inl 6737
This theorem is referenced by:  djune  6767  updjudhcoinlf  6769
  Copyright terms: Public domain W3C validator