Theorem 2ndinr 9845

Description: The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.)

Assertion

Ref	Expression
2ndinr	⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)

Proof of Theorem 2ndinr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-inr 9818	. . . 4 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2		opeq2 4805	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
3		elex 3452	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
4		opex 5403	. . . . 5 ⊢ ⟨1o, 𝑋⟩ ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨1o, 𝑋⟩ ∈ V)
6	1, 2, 3, 5	fvmptd3 6959	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inr‘𝑋) = ⟨1o, 𝑋⟩)
7	6	fveq2d 6831	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = (2nd ‘⟨1o, 𝑋⟩))
8		1oex 8405	. . 3 ⊢ 1o ∈ V
9		op2ndg 7944	. . 3 ⊢ ((1o ∈ V ∧ 𝑋 ∈ 𝑉) → (2nd ‘⟨1o, 𝑋⟩) = 𝑋)
10	8, 9	mpan 696	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘⟨1o, 𝑋⟩) = 𝑋)
11	7, 10	eqtrd 2774	1 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ⟨cop 4561  ‘cfv 6485  2^nd c2nd 7930  1_oc1o 8388  inrcinr 9815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-suc 6316  df-iota 6441  df-fun 6487  df-fv 6493  df-2nd 7932  df-1o 8395  df-inr 9818

This theorem is referenced by:  updjudhcoinrg  9848