Theorem 1stinr 9937

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 9911	. . . 4 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2		opeq2 4837	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
3		elex 3474	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
4		opex 5443	. . . . 5 ⊢ ⟨1o, 𝑋⟩ ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨1o, 𝑋⟩ ∈ V)
6	1, 2, 3, 5	fvmptd3 7014	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inr‘𝑋) = ⟨1o, 𝑋⟩)
7	6	fveq2d 6886	. 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = (1st ‘⟨1o, 𝑋⟩))
8		1oex 8468	. . 3 ⊢ 1o ∈ V
9		op1stg 8001	. . 3 ⊢ ((1o ∈ V ∧ 𝑋 ∈ 𝑉) → (1st ‘⟨1o, 𝑋⟩) = 1o)
10	8, 9	mpan 703	. 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 1570   ∈ wcel 2145  Vcvv 3453  ⟨cop 4593  ‘cfv 6537  1^st c1st 7987  1_oc1o 8451  inrcinr 9908

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

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 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 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-suc 6367  df-iota 6493  df-fun 6539  df-fv 6545  df-1st 7989  df-1o 8458  df-inr 9911

This theorem is used by:  updjudhcoinrg  9941