Theorem 1stinr 9687

Description: The first component of the value of a right injection is 1_o. (Contributed by AV, 27-Jun-2022.)

Assertion

Ref	Expression
1stinr	⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = 1o)

Proof of Theorem 1stinr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-inr 9661	. . . 4 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2		opeq2 4805	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
3		elex 3450	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
4		opex 5379	. . . . 5 ⊢ ⟨1o, 𝑋⟩ ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨1o, 𝑋⟩ ∈ V)
6	1, 2, 3, 5	fvmptd3 6898	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inr‘𝑋) = ⟨1o, 𝑋⟩)
7	6	fveq2d 6778	. 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = (1st ‘⟨1o, 𝑋⟩))
8		1oex 8307	. . 3 ⊢ 1o ∈ V
9		op1stg 7843	. . 3 ⊢ ((1o ∈ V ∧ 𝑋 ∈ 𝑉) → (1st ‘⟨1o, 𝑋⟩) = 1o)
10	8, 9	mpan 687	. 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘⟨1o, 𝑋⟩) = 1o)
11	7, 10	eqtrd 2778	1 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = 1o)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567  ‘cfv 6433  1^st c1st 7829  1_oc1o 8290  inrcinr 9658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-suc 6272  df-iota 6391  df-fun 6435  df-fv 6441  df-1st 7831  df-1o 8297  df-inr 9661

This theorem is referenced by:  updjudhcoinrg  9691