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Mirrors > Home > MPE Home > Th. List > efgrcl | Structured version Visualization version GIF version |
Description: Lemma for efgval 18488. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
Ref | Expression |
---|---|
efgval.w | ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) |
Ref | Expression |
---|---|
efgrcl | ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2on0 7841 | . . . 4 ⊢ 2o ≠ ∅ | |
2 | dmxp 5580 | . . . 4 ⊢ (2o ≠ ∅ → dom (𝐼 × 2o) = 𝐼) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ dom (𝐼 × 2o) = 𝐼 |
4 | elfvex 6471 | . . . . . 6 ⊢ (𝐴 ∈ ( I ‘Word (𝐼 × 2o)) → Word (𝐼 × 2o) ∈ V) | |
5 | efgval.w | . . . . . 6 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
6 | 4, 5 | eleq2s 2924 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → Word (𝐼 × 2o) ∈ V) |
7 | wrdexb 13592 | . . . . 5 ⊢ ((𝐼 × 2o) ∈ V ↔ Word (𝐼 × 2o) ∈ V) | |
8 | 6, 7 | sylibr 226 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐼 × 2o) ∈ V) |
9 | 8 | dmexd 7365 | . . 3 ⊢ (𝐴 ∈ 𝑊 → dom (𝐼 × 2o) ∈ V) |
10 | 3, 9 | syl5eqelr 2911 | . 2 ⊢ (𝐴 ∈ 𝑊 → 𝐼 ∈ V) |
11 | fvi 6506 | . . . 4 ⊢ (Word (𝐼 × 2o) ∈ V → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o)) | |
12 | 6, 11 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑊 → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o)) |
13 | 5, 12 | syl5eq 2873 | . 2 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o)) |
14 | 10, 13 | jca 507 | 1 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 Vcvv 3414 ∅c0 4146 I cid 5251 × cxp 5344 dom cdm 5346 ‘cfv 6127 2oc2o 7825 Word cword 13581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-2o 7832 df-er 8014 df-map 8129 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-fzo 12768 df-word 13582 |
This theorem is referenced by: efglem 18487 efgval 18488 efgtf 18493 efginvrel2 18498 efginvrel1 18499 efgredlemc 18517 efgcpbllemb 18528 efgcpbl2 18530 frgpcpbl 18532 frgpeccl 18534 frgpadd 18536 frgpinv 18537 frgpuplem 18545 frgpup1 18548 frgpnabllem1 18636 |
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