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Theorem 2ndinr 9035
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 9009 . . . . 5 inr = (𝑥 ∈ V ↦ ⟨1𝑜, 𝑥⟩)
21a1i 11 . . . 4 (𝑋𝑉 → inr = (𝑥 ∈ V ↦ ⟨1𝑜, 𝑥⟩))
3 opeq2 4596 . . . . 5 (𝑥 = 𝑋 → ⟨1𝑜, 𝑥⟩ = ⟨1𝑜, 𝑋⟩)
43adantl 469 . . . 4 ((𝑋𝑉𝑥 = 𝑋) → ⟨1𝑜, 𝑥⟩ = ⟨1𝑜, 𝑋⟩)
5 elex 3406 . . . 4 (𝑋𝑉𝑋 ∈ V)
6 opex 5122 . . . . 5 ⟨1𝑜, 𝑋⟩ ∈ V
76a1i 11 . . . 4 (𝑋𝑉 → ⟨1𝑜, 𝑋⟩ ∈ V)
82, 4, 5, 7fvmptd 6505 . . 3 (𝑋𝑉 → (inr‘𝑋) = ⟨1𝑜, 𝑋⟩)
98fveq2d 6408 . 2 (𝑋𝑉 → (2nd ‘(inr‘𝑋)) = (2nd ‘⟨1𝑜, 𝑋⟩))
10 1oex 7800 . . 3 1𝑜 ∈ V
11 op2ndg 7407 . . 3 ((1𝑜 ∈ V ∧ 𝑋𝑉) → (2nd ‘⟨1𝑜, 𝑋⟩) = 𝑋)
1210, 11mpan 673 . 2 (𝑋𝑉 → (2nd ‘⟨1𝑜, 𝑋⟩) = 𝑋)
139, 12eqtrd 2840 1 (𝑋𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2156  Vcvv 3391  cop 4376  cmpt 4923  cfv 6097  2nd c2nd 7393  1𝑜c1o 7785  inrcinr 9006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-ord 5939  df-on 5940  df-suc 5942  df-iota 6060  df-fun 6099  df-fv 6105  df-2nd 7395  df-1o 7792  df-inr 9009
This theorem is referenced by:  updjudhcoinrg  9038
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