Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > efgmf | Structured version Visualization version GIF version |
Description: The formal inverse operation is an endofunction on the generating set. (Contributed by Mario Carneiro, 27-Sep-2015.) |
Ref | Expression |
---|---|
efgmval.m | ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ 〈𝑦, (1𝑜 ∖ 𝑧)〉) |
Ref | Expression |
---|---|
efgmf | ⊢ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2oconcl 7470 | . . . 4 ⊢ (𝑧 ∈ 2𝑜 → (1𝑜 ∖ 𝑧) ∈ 2𝑜) | |
2 | opelxpi 5072 | . . . 4 ⊢ ((𝑦 ∈ 𝐼 ∧ (1𝑜 ∖ 𝑧) ∈ 2𝑜) → 〈𝑦, (1𝑜 ∖ 𝑧)〉 ∈ (𝐼 × 2𝑜)) | |
3 | 1, 2 | sylan2 490 | . . 3 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜) → 〈𝑦, (1𝑜 ∖ 𝑧)〉 ∈ (𝐼 × 2𝑜)) |
4 | 3 | rgen2 2958 | . 2 ⊢ ∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 〈𝑦, (1𝑜 ∖ 𝑧)〉 ∈ (𝐼 × 2𝑜) |
5 | efgmval.m | . . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ 〈𝑦, (1𝑜 ∖ 𝑧)〉) | |
6 | 5 | fmpt2 7126 | . 2 ⊢ (∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 〈𝑦, (1𝑜 ∖ 𝑧)〉 ∈ (𝐼 × 2𝑜) ↔ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) |
7 | 4, 6 | mpbi 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 |
Copyright terms: Public domain | W3C validator |