Theorem 2ndinr 6970

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 6941	. . . . 5 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2	1	a1i 9	. . . 4 ⊢ (𝑋 ∈ 𝑉 → inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩))
3		opeq2 3714	. . . . 5 ⊢ (𝑥 = 𝑋 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
4	3	adantl 275	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
5		elex 2700	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
6		1on 6328	. . . . 5 ⊢ 1o ∈ On
7		opexg 4158	. . . . 5 ⊢ ((1o ∈ On ∧ 𝑋 ∈ 𝑉) → ⟨1o, 𝑋⟩ ∈ V)
8	6, 7	mpan 421	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨1o, 𝑋⟩ ∈ V)
9	2, 4, 5, 8	fvmptd 5510	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inr‘𝑋) = ⟨1o, 𝑋⟩)
10	9	fveq2d 5433	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = (2nd ‘⟨1o, 𝑋⟩))
11		op2ndg 6057	. . 3 ⊢ ((1o ∈ On ∧ 𝑋 ∈ 𝑉) → (2nd ‘⟨1o, 𝑋⟩) = 𝑋)
12	6, 11	mpan 421	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘⟨1o, 𝑋⟩) = 𝑋)
13	10, 12	eqtrd 2173	1 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  Vcvv 2689  ⟨cop 3535   ↦ cmpt 3997  Oncon0 4293  ‘cfv 5131  2^nd c2nd 6045  1_oc1o 6314  inrcinr 6939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fv 5139  df-2nd 6047  df-1o 6321  df-inr 6941

This theorem is referenced by:  updjudhcoinrg  6974