Theorem 1stinl 9946

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	⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅)

Proof of Theorem 1stinl

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-inl 9921	. . . 4 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2		opeq2 4855	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
3		elex 3485	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
4		opex 5444	. . . . 5 ⊢ ⟨∅, 𝑋⟩ ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨∅, 𝑋⟩ ∈ V)
6	1, 2, 3, 5	fvmptd3 7014	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
7	6	fveq2d 6885	. 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = (1st ‘⟨∅, 𝑋⟩))
8		0ex 5282	. . 3 ⊢ ∅ ∈ V
9		op1stg 8005	. . 3 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → (1st ‘⟨∅, 𝑋⟩) = ∅)
10	8, 9	mpan 690	. 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘⟨∅, 𝑋⟩) = ∅)
11	7, 10	eqtrd 2771	1 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3464  ∅c0 4313  ⟨cop 4612  ‘cfv 6536  1^st c1st 7991  inlcinl 9918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fv 6544  df-1st 7993  df-inl 9921

This theorem is referenced by:  updjudhcoinlf  9951