Theorem 1stinl 7075

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 7048	. . . . 5 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2	1	a1i 9	. . . 4 ⊢ (𝑋 ∈ 𝑉 → inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩))
3		opeq2 3781	. . . . 5 ⊢ (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
4	3	adantl 277	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
5		elex 2750	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
6		0ex 4132	. . . . 5 ⊢ ∅ ∈ V
7		opexg 4230	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → ⟨∅, 𝑋⟩ ∈ V)
8	6, 7	mpan 424	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨∅, 𝑋⟩ ∈ V)
9	2, 4, 5, 8	fvmptd 5599	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
10	9	fveq2d 5521	. 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = (1st ‘⟨∅, 𝑋⟩))
11		op1stg 6153	. . 3 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → (1st ‘⟨∅, 𝑋⟩) = ∅)
12	6, 11	mpan 424	. 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘⟨∅, 𝑋⟩) = ∅)
13	10, 12	eqtrd 2210	1 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅)

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  Vcvv 2739  ∅c0 3424  ⟨cop 3597   ↦ cmpt 4066  ‘cfv 5218  1^st c1st 6141  inlcinl 7046

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fv 5226  df-1st 6143  df-inl 7048

This theorem is referenced by:  djune  7079  updjudhcoinlf  7081  subctctexmid  14835