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Theorem 1stinr 7072
Description: The first component of the value of a right injection is 1o. (Contributed by AV, 27-Jun-2022.)
Assertion
Ref Expression
1stinr (𝑋𝑉 → (1st ‘(inr‘𝑋)) = 1o)

Proof of Theorem 1stinr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-inr 7044 . . . . 5 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
21a1i 9 . . . 4 (𝑋𝑉 → inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩))
3 opeq2 3779 . . . . 5 (𝑥 = 𝑋 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
43adantl 277 . . . 4 ((𝑋𝑉𝑥 = 𝑋) → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
5 elex 2748 . . . 4 (𝑋𝑉𝑋 ∈ V)
6 1on 6421 . . . . 5 1o ∈ On
7 opexg 4227 . . . . 5 ((1o ∈ On ∧ 𝑋𝑉) → ⟨1o, 𝑋⟩ ∈ V)
86, 7mpan 424 . . . 4 (𝑋𝑉 → ⟨1o, 𝑋⟩ ∈ V)
92, 4, 5, 8fvmptd 5596 . . 3 (𝑋𝑉 → (inr‘𝑋) = ⟨1o, 𝑋⟩)
109fveq2d 5518 . 2 (𝑋𝑉 → (1st ‘(inr‘𝑋)) = (1st ‘⟨1o, 𝑋⟩))
11 op1stg 6148 . . 3 ((1o ∈ On ∧ 𝑋𝑉) → (1st ‘⟨1o, 𝑋⟩) = 1o)
126, 11mpan 424 . 2 (𝑋𝑉 → (1st ‘⟨1o, 𝑋⟩) = 1o)
1310, 12eqtrd 2210 1 (𝑋𝑉 → (1st ‘(inr‘𝑋)) = 1o)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  Vcvv 2737  cop 3595  cmpt 4063  Oncon0 4362  cfv 5215  1st c1st 6136  1oc1o 6407  inrcinr 7042
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-iord 4365  df-on 4367  df-suc 4370  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-iota 5177  df-fun 5217  df-fv 5223  df-1st 6138  df-1o 6414  df-inr 7044
This theorem is referenced by:  djune  7074  updjudhcoinrg  7077
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