Theorem 1stinl 7365

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 7338	. . . . 5 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2	1	a1i 9	. . . 4 ⊢ (𝑋 ∈ 𝑉 → inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩))
3		opeq2 3884	. . . . 5 ⊢ (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
4	3	adantl 277	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
5		elex 2825	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
6		0ex 4237	. . . . 5 ⊢ ∅ ∈ V
7		opexg 4344	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → ⟨∅, 𝑋⟩ ∈ V)
8	6, 7	mpan 424	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨∅, 𝑋⟩ ∈ V)
9	2, 4, 5, 8	fvmptd 5758	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
10	9	fveq2d 5674	. 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = (1st ‘⟨∅, 𝑋⟩))
11		op1stg 6344	. . 3 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → (1st ‘⟨∅, 𝑋⟩) = ∅)
12	6, 11	mpan 424	. 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘⟨∅, 𝑋⟩) = ∅)
13	10, 12	eqtrd 2265	1 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  Vcvv 2813  ∅c0 3508  ⟨cop 3692   ↦ cmpt 4171  ‘cfv 5352  1^st c1st 6332  inlcinl 7336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fv 5360  df-1st 6334  df-inl 7338

This theorem is referenced by:  djune  7369  updjudhcoinlf  7371  subctctexmid  16774