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Theorem 2ndinr 8956
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 8930 . . . . 5 inr = (𝑥 ∈ V ↦ ⟨1𝑜, 𝑥⟩)
21a1i 11 . . . 4 (𝑋𝑉 → inr = (𝑥 ∈ V ↦ ⟨1𝑜, 𝑥⟩))
3 opeq2 4540 . . . . 5 (𝑥 = 𝑋 → ⟨1𝑜, 𝑥⟩ = ⟨1𝑜, 𝑋⟩)
43adantl 467 . . . 4 ((𝑋𝑉𝑥 = 𝑋) → ⟨1𝑜, 𝑥⟩ = ⟨1𝑜, 𝑋⟩)
5 elex 3364 . . . 4 (𝑋𝑉𝑋 ∈ V)
6 opex 5060 . . . . 5 ⟨1𝑜, 𝑋⟩ ∈ V
76a1i 11 . . . 4 (𝑋𝑉 → ⟨1𝑜, 𝑋⟩ ∈ V)
82, 4, 5, 7fvmptd 6430 . . 3 (𝑋𝑉 → (inr‘𝑋) = ⟨1𝑜, 𝑋⟩)
98fveq2d 6336 . 2 (𝑋𝑉 → (2nd ‘(inr‘𝑋)) = (2nd ‘⟨1𝑜, 𝑋⟩))
10 1oex 7721 . . 3 1𝑜 ∈ V
11 op2ndg 7328 . . 3 ((1𝑜 ∈ V ∧ 𝑋𝑉) → (2nd ‘⟨1𝑜, 𝑋⟩) = 𝑋)
1210, 11mpan 670 . 2 (𝑋𝑉 → (2nd ‘⟨1𝑜, 𝑋⟩) = 𝑋)
139, 12eqtrd 2805 1 (𝑋𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  Vcvv 3351  cop 4322  cmpt 4863  cfv 6031  2nd c2nd 7314  1𝑜c1o 7706  inrcinr 8927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fv 6039  df-2nd 7316  df-1o 7713  df-inr 8930
This theorem is referenced by:  updjudhcoinrg  8959
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