Theorem 2ndinr 7143

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 7114	. . . . 5 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2	1	a1i 9	. . . 4 ⊢ (𝑋 ∈ 𝑉 → inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩))
3		opeq2 3809	. . . . 5 ⊢ (𝑥 = 𝑋 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
4	3	adantl 277	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
5		elex 2774	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
6		1on 6481	. . . . 5 ⊢ 1o ∈ On
7		opexg 4261	. . . . 5 ⊢ ((1o ∈ On ∧ 𝑋 ∈ 𝑉) → ⟨1o, 𝑋⟩ ∈ V)
8	6, 7	mpan 424	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨1o, 𝑋⟩ ∈ V)
9	2, 4, 5, 8	fvmptd 5642	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inr‘𝑋) = ⟨1o, 𝑋⟩)
10	9	fveq2d 5562	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = (2nd ‘⟨1o, 𝑋⟩))
11		op2ndg 6209	. . 3 ⊢ ((1o ∈ On ∧ 𝑋 ∈ 𝑉) → (2nd ‘⟨1o, 𝑋⟩) = 𝑋)
12	6, 11	mpan 424	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘⟨1o, 𝑋⟩) = 𝑋)
13	10, 12	eqtrd 2229	1 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  Vcvv 2763  ⟨cop 3625   ↦ cmpt 4094  Oncon0 4398  ‘cfv 5258  2^nd c2nd 6197  1_oc1o 6467  inrcinr 7112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fv 5266  df-2nd 6199  df-1o 6474  df-inr 7114

This theorem is referenced by:  updjudhcoinrg  7147