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Theorem 1stinl 7051
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 7024 . . . . 5 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
21a1i 9 . . . 4 (𝑋𝑉 → inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩))
3 opeq2 3766 . . . . 5 (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
43adantl 275 . . . 4 ((𝑋𝑉𝑥 = 𝑋) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
5 elex 2741 . . . 4 (𝑋𝑉𝑋 ∈ V)
6 0ex 4116 . . . . 5 ∅ ∈ V
7 opexg 4213 . . . . 5 ((∅ ∈ V ∧ 𝑋𝑉) → ⟨∅, 𝑋⟩ ∈ V)
86, 7mpan 422 . . . 4 (𝑋𝑉 → ⟨∅, 𝑋⟩ ∈ V)
92, 4, 5, 8fvmptd 5577 . . 3 (𝑋𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
109fveq2d 5500 . 2 (𝑋𝑉 → (1st ‘(inl‘𝑋)) = (1st ‘⟨∅, 𝑋⟩))
11 op1stg 6129 . . 3 ((∅ ∈ V ∧ 𝑋𝑉) → (1st ‘⟨∅, 𝑋⟩) = ∅)
126, 11mpan 422 . 2 (𝑋𝑉 → (1st ‘⟨∅, 𝑋⟩) = ∅)
1310, 12eqtrd 2203 1 (𝑋𝑉 → (1st ‘(inl‘𝑋)) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wcel 2141  Vcvv 2730  c0 3414  cop 3586  cmpt 4050  cfv 5198  1st c1st 6117  inlcinl 7022
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 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418
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 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-1st 6119  df-inl 7024
This theorem is referenced by:  djune  7055  updjudhcoinlf  7057  subctctexmid  14034
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