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Theorem efgmf 19746
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 8540 . . . 4 (𝑧 ∈ 2o → (1o𝑧) ∈ 2o)
2 opelxpi 5726 . . . 4 ((𝑦𝐼 ∧ (1o𝑧) ∈ 2o) → ⟨𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o))
31, 2sylan2 593 . . 3 ((𝑦𝐼𝑧 ∈ 2o) → ⟨𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o))
43rgen2 3197 . 2 𝑦𝐼𝑧 ∈ 2o𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o)
5 efgmval.m . . 3 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
65fmpo 8092 . 2 (∀𝑦𝐼𝑧 ∈ 2o𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o) ↔ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o))
74, 6mpbi 230 1 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  wral 3059  cdif 3960  cop 4637   × cxp 5687  wf 6559  cmpo 7433  1oc1o 8498  2oc2o 8499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506
This theorem is referenced by:  efgtf  19755  efgtlen  19759  efginvrel2  19760  efginvrel1  19761  efgredleme  19776  efgredlemc  19778  efgcpbllemb  19788  frgp0  19793  frgpinv  19797  vrgpinv  19802  frgpnabllem1  19906
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