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Theorem 2ndinr 6962
 Description: The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.)
Assertion
Ref Expression
2ndinr (𝑋𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)

Proof of Theorem 2ndinr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-inr 6933 . . . . 5 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
21a1i 9 . . . 4 (𝑋𝑉 → inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩))
3 opeq2 3706 . . . . 5 (𝑥 = 𝑋 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
43adantl 275 . . . 4 ((𝑋𝑉𝑥 = 𝑋) → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
5 elex 2697 . . . 4 (𝑋𝑉𝑋 ∈ V)
6 1on 6320 . . . . 5 1o ∈ On
7 opexg 4150 . . . . 5 ((1o ∈ On ∧ 𝑋𝑉) → ⟨1o, 𝑋⟩ ∈ V)
86, 7mpan 420 . . . 4 (𝑋𝑉 → ⟨1o, 𝑋⟩ ∈ V)
92, 4, 5, 8fvmptd 5502 . . 3 (𝑋𝑉 → (inr‘𝑋) = ⟨1o, 𝑋⟩)
109fveq2d 5425 . 2 (𝑋𝑉 → (2nd ‘(inr‘𝑋)) = (2nd ‘⟨1o, 𝑋⟩))
11 op2ndg 6049 . . 3 ((1o ∈ On ∧ 𝑋𝑉) → (2nd ‘⟨1o, 𝑋⟩) = 𝑋)
126, 11mpan 420 . 2 (𝑋𝑉 → (2nd ‘⟨1o, 𝑋⟩) = 𝑋)
1310, 12eqtrd 2172 1 (𝑋𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  Vcvv 2686  ⟨cop 3530   ↦ cmpt 3989  Oncon0 4285  ‘cfv 5123  2nd c2nd 6037  1oc1o 6306  inrcinr 6931 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-2nd 6039  df-1o 6313  df-inr 6933 This theorem is referenced by:  updjudhcoinrg  6966
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