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Theorem efgmf 17949
 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 7470 . . . 4 (𝑧 ∈ 2𝑜 → (1𝑜𝑧) ∈ 2𝑜)
2 opelxpi 5072 . . . 4 ((𝑦𝐼 ∧ (1𝑜𝑧) ∈ 2𝑜) → ⟨𝑦, (1𝑜𝑧)⟩ ∈ (𝐼 × 2𝑜))
31, 2sylan2 490 . . 3 ((𝑦𝐼𝑧 ∈ 2𝑜) → ⟨𝑦, (1𝑜𝑧)⟩ ∈ (𝐼 × 2𝑜))
43rgen2 2958 . 2 𝑦𝐼𝑧 ∈ 2𝑜𝑦, (1𝑜𝑧)⟩ ∈ (𝐼 × 2𝑜)
5 efgmval.m . . 3 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
65fmpt2 7126 . 2 (∀𝑦𝐼𝑧 ∈ 2𝑜𝑦, (1𝑜𝑧)⟩ ∈ (𝐼 × 2𝑜) ↔ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜))
74, 6mpbi 219 1 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537  ⟨cop 4131   × cxp 5036  ⟶wf 5800   ↦ cmpt2 6551  1𝑜c1o 7440  2𝑜c2o 7441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-1o 7447  df-2o 7448 This theorem is referenced by:  efgtf  17958  efgtlen  17962  efginvrel2  17963  efginvrel1  17964  efgredleme  17979  efgredlemc  17981  efgcpbllemb  17991  frgp0  17996  frgpinv  18000  vrgpinv  18005  frgpnabllem1  18099
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