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Theorem 1stinl 7018
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 6991 . . . . 5 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
21a1i 9 . . . 4 (𝑋𝑉 → inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩))
3 opeq2 3742 . . . . 5 (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
43adantl 275 . . . 4 ((𝑋𝑉𝑥 = 𝑋) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
5 elex 2723 . . . 4 (𝑋𝑉𝑋 ∈ V)
6 0ex 4091 . . . . 5 ∅ ∈ V
7 opexg 4188 . . . . 5 ((∅ ∈ V ∧ 𝑋𝑉) → ⟨∅, 𝑋⟩ ∈ V)
86, 7mpan 421 . . . 4 (𝑋𝑉 → ⟨∅, 𝑋⟩ ∈ V)
92, 4, 5, 8fvmptd 5549 . . 3 (𝑋𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
109fveq2d 5472 . 2 (𝑋𝑉 → (1st ‘(inl‘𝑋)) = (1st ‘⟨∅, 𝑋⟩))
11 op1stg 6098 . . 3 ((∅ ∈ V ∧ 𝑋𝑉) → (1st ‘⟨∅, 𝑋⟩) = ∅)
126, 11mpan 421 . 2 (𝑋𝑉 → (1st ‘⟨∅, 𝑋⟩) = ∅)
1310, 12eqtrd 2190 1 (𝑋𝑉 → (1st ‘(inl‘𝑋)) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335  wcel 2128  Vcvv 2712  c0 3394  cop 3563  cmpt 4025  cfv 5170  1st c1st 6086  inlcinl 6989
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-iota 5135  df-fun 5172  df-fv 5178  df-1st 6088  df-inl 6991
This theorem is referenced by:  djune  7022  updjudhcoinlf  7024  subctctexmid  13573
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