Theorem 1stinr 9922

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 9896	. . . 4 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2		opeq2 4838	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
3		elex 3475	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
4		opex 5444	. . . . 5 ⊢ ⟨1o, 𝑋⟩ ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨1o, 𝑋⟩ ∈ V)
6	1, 2, 3, 5	fvmptd3 7013	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inr‘𝑋) = ⟨1o, 𝑋⟩)
7	6	fveq2d 6885	. 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = (1st ‘⟨1o, 𝑋⟩))
8		1oex 8461	. . 3 ⊢ 1o ∈ V
9		op1stg 7996	. . 3 ⊢ ((1o ∈ V ∧ 𝑋 ∈ 𝑉) → (1st ‘⟨1o, 𝑋⟩) = 1o)
10	8, 9	mpan 702	. 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘⟨1o, 𝑋⟩) = 1o)
11	7, 10	eqtrd 2797	1 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = 1o)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  Vcvv 3454  ⟨cop 4594  ‘cfv 6536  1^st c1st 7982  1_oc1o 8444  inrcinr 9893

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-suc 6366  df-iota 6492  df-fun 6538  df-fv 6544  df-1st 7984  df-1o 8451  df-inr 9896

This theorem is used by:  updjudhcoinrg  9926