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Theorem efgmf 18839
 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 8118 . . . 4 (𝑧 ∈ 2o → (1o𝑧) ∈ 2o)
2 opelxpi 5557 . . . 4 ((𝑦𝐼 ∧ (1o𝑧) ∈ 2o) → ⟨𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o))
31, 2sylan2 595 . . 3 ((𝑦𝐼𝑧 ∈ 2o) → ⟨𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o))
43rgen2 3168 . 2 𝑦𝐼𝑧 ∈ 2o𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o)
5 efgmval.m . . 3 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
65fmpo 7755 . 2 (∀𝑦𝐼𝑧 ∈ 2o𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o) ↔ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o))
74, 6mpbi 233 1 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∖ cdif 3878  ⟨cop 4531   × cxp 5518  ⟶wf 6323   ∈ cmpo 7142  1oc1o 8085  2oc2o 8086 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-ord 6165  df-on 6166  df-suc 6168  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-fv 6335  df-oprab 7144  df-mpo 7145  df-1st 7678  df-2nd 7679  df-1o 8092  df-2o 8093 This theorem is referenced by:  efgtf  18848  efgtlen  18852  efginvrel2  18853  efginvrel1  18854  efgredleme  18869  efgredlemc  18871  efgcpbllemb  18881  frgp0  18886  frgpinv  18890  vrgpinv  18895  frgpnabllem1  18994
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