Theorem 1stinl 9929

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 9904	. . . 4 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2		opeq2 4841	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
3		elex 3478	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
4		opex 5447	. . . . 5 ⊢ ⟨∅, 𝑋⟩ ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨∅, 𝑋⟩ ∈ V)
6	1, 2, 3, 5	fvmptd3 7017	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
7	6	fveq2d 6889	. 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = (1st ‘⟨∅, 𝑋⟩))
8		0ex 5272	. . 3 ⊢ ∅ ∈ V
9		op1stg 8004	. . 3 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → (1st ‘⟨∅, 𝑋⟩) = ∅)
10	8, 9	mpan 703	. 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘⟨∅, 𝑋⟩) = ∅)
11	7, 10	eqtrd 2800	1 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ∅c0 4286  ⟨cop 4597  ‘cfv 6540  1^st c1st 7990  inlcinl 9901

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fv 6548  df-1st 7992  df-inl 9904

This theorem is used by:  updjudhcoinlf  9934