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Theorem djurf1o 9907
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 9897 . . 3 inr = (š‘„ ∈ V ↦ ⟨1o, š‘„āŸ©)
2 1onn 8638 . . . . . 6 1o ∈ ω
3 snidg 4657 . . . . . 6 (1o ∈ ω → 1o ∈ {1o})
42, 3ax-mp 5 . . . . 5 1o ∈ {1o}
5 opelxpi 5706 . . . . 5 ((1o ∈ {1o} ∧ š‘„ ∈ V) → ⟨1o, š‘„āŸ© ∈ ({1o} Ɨ V))
64, 5mpan 687 . . . 4 (š‘„ ∈ V → ⟨1o, š‘„āŸ© ∈ ({1o} Ɨ V))
76adantl 481 . . 3 ((⊤ ∧ š‘„ ∈ V) → ⟨1o, š‘„āŸ© ∈ ({1o} Ɨ V))
8 fvexd 6899 . . 3 ((⊤ ∧ š‘¦ ∈ ({1o} Ɨ V)) → (2nd ā€˜š‘¦) ∈ V)
9 1st2nd2 8010 . . . . . . . 8 (š‘¦ ∈ ({1o} Ɨ V) → š‘¦ = ⟨(1st ā€˜š‘¦), (2nd ā€˜š‘¦)⟩)
10 xp1st 8003 . . . . . . . . . 10 (š‘¦ ∈ ({1o} Ɨ V) → (1st ā€˜š‘¦) ∈ {1o})
11 elsni 4640 . . . . . . . . . 10 ((1st ā€˜š‘¦) ∈ {1o} → (1st ā€˜š‘¦) = 1o)
1210, 11syl 17 . . . . . . . . 9 (š‘¦ ∈ ({1o} Ɨ V) → (1st ā€˜š‘¦) = 1o)
1312opeq1d 4874 . . . . . . . 8 (š‘¦ ∈ ({1o} Ɨ V) → ⟨(1st ā€˜š‘¦), (2nd ā€˜š‘¦)⟩ = ⟨1o, (2nd ā€˜š‘¦)⟩)
149, 13eqtrd 2766 . . . . . . 7 (š‘¦ ∈ ({1o} Ɨ V) → š‘¦ = ⟨1o, (2nd ā€˜š‘¦)⟩)
1514eqeq2d 2737 . . . . . 6 (š‘¦ ∈ ({1o} Ɨ V) → (⟨1o, š‘„āŸ© = š‘¦ ↔ ⟨1o, š‘„āŸ© = ⟨1o, (2nd ā€˜š‘¦)⟩))
16 eqcom 2733 . . . . . 6 (⟨1o, š‘„āŸ© = š‘¦ ↔ š‘¦ = ⟨1o, š‘„āŸ©)
17 eqid 2726 . . . . . . 7 1o = 1o
18 1oex 8474 . . . . . . . 8 1o ∈ V
19 vex 3472 . . . . . . . 8 š‘„ ∈ V
2018, 19opth 5469 . . . . . . 7 (⟨1o, š‘„āŸ© = ⟨1o, (2nd ā€˜š‘¦)⟩ ↔ (1o = 1o ∧ š‘„ = (2nd ā€˜š‘¦)))
2117, 20mpbiran 706 . . . . . 6 (⟨1o, š‘„āŸ© = ⟨1o, (2nd ā€˜š‘¦)⟩ ↔ š‘„ = (2nd ā€˜š‘¦))
2215, 16, 213bitr3g 313 . . . . 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 726 . . 3 ((⊤ ∧ (š‘„ ∈ V ∧ š‘¦ ∈ ({1o} Ɨ V))) → (š‘„ = (2nd ā€˜š‘¦) ↔ š‘¦ = ⟨1o, š‘„āŸ©))
251, 7, 8, 24f1o2d 7656 . 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 395   = wceq 1533  āФwtru 1534   ∈ wcel 2098  Vcvv 3468  {csn 4623  āŸØcop 4629   Ɨ cxp 5667  ā€“1-1-onto→wf1o 6535  ā€˜cfv 6536  Ļ‰com 7851  1st c1st 7969  2nd c2nd 7970  1oc1o 8457  inrcinr 9894
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7852  df-1st 7971  df-2nd 7972  df-1o 8464  df-inr 9897
This theorem is referenced by:  inrresf  9910  inrresf1  9911  djuin  9912  djuun  9920
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