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Theorem efgmf 18050
Description: The formal inverse operation is an endofunction on the generating set. (Contributed by Mario Carneiro, 27-Sep-2015.)
Hypothesis
Ref Expression
efgmval.m 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
Assertion
Ref Expression
efgmf 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
Distinct variable group:   𝑦,𝑧,𝐼
Allowed substitution hints:   𝑀(𝑦,𝑧)

Proof of Theorem efgmf
StepHypRef Expression
1 2oconcl 7531 . . . 4 (𝑧 ∈ 2𝑜 → (1𝑜𝑧) ∈ 2𝑜)
2 opelxpi 5110 . . . 4 ((𝑦𝐼 ∧ (1𝑜𝑧) ∈ 2𝑜) → ⟨𝑦, (1𝑜𝑧)⟩ ∈ (𝐼 × 2𝑜))
31, 2sylan2 491 . . 3 ((𝑦𝐼𝑧 ∈ 2𝑜) → ⟨𝑦, (1𝑜𝑧)⟩ ∈ (𝐼 × 2𝑜))
43rgen2 2969 . 2 𝑦𝐼𝑧 ∈ 2𝑜𝑦, (1𝑜𝑧)⟩ ∈ (𝐼 × 2𝑜)
5 efgmval.m . . 3 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
65fmpt2 7185 . 2 (∀𝑦𝐼𝑧 ∈ 2𝑜𝑦, (1𝑜𝑧)⟩ ∈ (𝐼 × 2𝑜) ↔ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜))
74, 6mpbi 220 1 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  wral 2907  cdif 3553  cop 4156   × cxp 5074  wf 5845  cmpt2 6609  1𝑜c1o 7501  2𝑜c2o 7502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-ord 5687  df-on 5688  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-fv 5857  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-1o 7508  df-2o 7509
This theorem is referenced by:  efgtf  18059  efgtlen  18063  efginvrel2  18064  efginvrel1  18065  efgredleme  18080  efgredlemc  18082  efgcpbllemb  18092  frgp0  18097  frgpinv  18101  vrgpinv  18106  frgpnabllem1  18200
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