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Theorem djurf1o 9944
Description: The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
Assertion
Ref Expression
djurf1o inr:V–1-1-onto→({1o} Ɨ V)

Proof of Theorem djurf1o
Dummy variables š‘„ š‘¦ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-inr 9934 . . 3 inr = (š‘„ ∈ V ↦ ⟨1o, š‘„āŸ©)
2 1onn 8667 . . . . . 6 1o ∈ ω
3 snidg 4667 . . . . . 6 (1o ∈ ω → 1o ∈ {1o})
42, 3ax-mp 5 . . . . 5 1o ∈ {1o}
5 opelxpi 5719 . . . . 5 ((1o ∈ {1o} ∧ š‘„ ∈ V) → ⟨1o, š‘„āŸ© ∈ ({1o} Ɨ V))
64, 5mpan 688 . . . 4 (š‘„ ∈ V → ⟨1o, š‘„āŸ© ∈ ({1o} Ɨ V))
76adantl 480 . . 3 ((⊤ ∧ š‘„ ∈ V) → ⟨1o, š‘„āŸ© ∈ ({1o} Ɨ V))
8 fvexd 6917 . . 3 ((⊤ ∧ š‘¦ ∈ ({1o} Ɨ V)) → (2nd ā€˜š‘¦) ∈ V)
9 1st2nd2 8038 . . . . . . . 8 (š‘¦ ∈ ({1o} Ɨ V) → š‘¦ = ⟨(1st ā€˜š‘¦), (2nd ā€˜š‘¦)⟩)
10 xp1st 8031 . . . . . . . . . 10 (š‘¦ ∈ ({1o} Ɨ V) → (1st ā€˜š‘¦) ∈ {1o})
11 elsni 4649 . . . . . . . . . 10 ((1st ā€˜š‘¦) ∈ {1o} → (1st ā€˜š‘¦) = 1o)
1210, 11syl 17 . . . . . . . . 9 (š‘¦ ∈ ({1o} Ɨ V) → (1st ā€˜š‘¦) = 1o)
1312opeq1d 4884 . . . . . . . 8 (š‘¦ ∈ ({1o} Ɨ V) → ⟨(1st ā€˜š‘¦), (2nd ā€˜š‘¦)⟩ = ⟨1o, (2nd ā€˜š‘¦)⟩)
149, 13eqtrd 2768 . . . . . . 7 (š‘¦ ∈ ({1o} Ɨ V) → š‘¦ = ⟨1o, (2nd ā€˜š‘¦)⟩)
1514eqeq2d 2739 . . . . . 6 (š‘¦ ∈ ({1o} Ɨ V) → (⟨1o, š‘„āŸ© = š‘¦ ↔ ⟨1o, š‘„āŸ© = ⟨1o, (2nd ā€˜š‘¦)⟩))
16 eqcom 2735 . . . . . 6 (⟨1o, š‘„āŸ© = š‘¦ ↔ š‘¦ = ⟨1o, š‘„āŸ©)
17 eqid 2728 . . . . . . 7 1o = 1o
18 1oex 8503 . . . . . . . 8 1o ∈ V
19 vex 3477 . . . . . . . 8 š‘„ ∈ V
2018, 19opth 5482 . . . . . . 7 (⟨1o, š‘„āŸ© = ⟨1o, (2nd ā€˜š‘¦)⟩ ↔ (1o = 1o ∧ š‘„ = (2nd ā€˜š‘¦)))
2117, 20mpbiran 707 . . . . . 6 (⟨1o, š‘„āŸ© = ⟨1o, (2nd ā€˜š‘¦)⟩ ↔ š‘„ = (2nd ā€˜š‘¦))
2215, 16, 213bitr3g 312 . . . . 5 (š‘¦ ∈ ({1o} Ɨ V) → (š‘¦ = ⟨1o, š‘„āŸ© ↔ š‘„ = (2nd ā€˜š‘¦)))
2322bicomd 222 . . . 4 (š‘¦ ∈ ({1o} Ɨ V) → (š‘„ = (2nd ā€˜š‘¦) ↔ š‘¦ = ⟨1o, š‘„āŸ©))
2423ad2antll 727 . . 3 ((⊤ ∧ (š‘„ ∈ V ∧ š‘¦ ∈ ({1o} Ɨ V))) → (š‘„ = (2nd ā€˜š‘¦) ↔ š‘¦ = ⟨1o, š‘„āŸ©))
251, 7, 8, 24f1o2d 7681 . 2 (⊤ → inr:V–1-1-onto→({1o} Ɨ V))
2625mptru 1540 1 inr:V–1-1-onto→({1o} Ɨ V)
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533  āФwtru 1534   ∈ wcel 2098  Vcvv 3473  {csn 4632  āŸØcop 4638   Ɨ cxp 5680  ā€“1-1-onto→wf1o 6552  ā€˜cfv 6553  Ļ‰com 7876  1st c1st 7997  2nd c2nd 7998  1oc1o 8486  inrcinr 9931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-om 7877  df-1st 7999  df-2nd 8000  df-1o 8493  df-inr 9934
This theorem is referenced by:  inrresf  9947  inrresf1  9948  djuin  9949  djuun  9957
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