Theorem 2ndinr 9888

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 9861	. . . 4 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2		opeq2 4832	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
3		elex 3475	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
4		opex 5431	. . . . 5 ⊢ ⟨1o, 𝑋⟩ ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨1o, 𝑋⟩ ∈ V)
6	1, 2, 3, 5	fvmptd3 6999	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inr‘𝑋) = ⟨1o, 𝑋⟩)
7	6	fveq2d 6871	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = (2nd ‘⟨1o, 𝑋⟩))
8		1oex 8447	. . 3 ⊢ 1o ∈ V
9		op2ndg 7983	. . 3 ⊢ ((1o ∈ V ∧ 𝑋 ∈ 𝑉) → (2nd ‘⟨1o, 𝑋⟩) = 𝑋)
10	8, 9	mpan 700	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘⟨1o, 𝑋⟩) = 𝑋)
11	7, 10	eqtrd 2797	1 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142  Vcvv 3454  ⟨cop 4588  ‘cfv 6521  2^nd c2nd 7969  1_oc1o 8430  inrcinr 9858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-suc 6352  df-iota 6477  df-fun 6523  df-fv 6529  df-2nd 7971  df-1o 8437  df-inr 9861

This theorem is referenced by:  updjudhcoinrg  9891