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Mirrors > Home > MPE Home > Th. List > efgrcl | Structured version Visualization version GIF version |
Description: Lemma for efgval 18442. (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 7810 | . . . 4 ⊢ 2𝑜 ≠ ∅ | |
2 | dmxp 5548 | . . . 4 ⊢ (2𝑜 ≠ ∅ → dom (𝐼 × 2𝑜) = 𝐼) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ dom (𝐼 × 2𝑜) = 𝐼 |
4 | elfvex 6446 | . . . . . 6 ⊢ (𝐴 ∈ ( I ‘Word (𝐼 × 2𝑜)) → Word (𝐼 × 2𝑜) ∈ V) | |
5 | efgval.w | . . . . . 6 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜)) | |
6 | 4, 5 | eleq2s 2897 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → Word (𝐼 × 2𝑜) ∈ V) |
7 | wrdexb 13544 | . . . . 5 ⊢ ((𝐼 × 2𝑜) ∈ V ↔ Word (𝐼 × 2𝑜) ∈ V) | |
8 | 6, 7 | sylibr 226 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐼 × 2𝑜) ∈ V) |
9 | 8 | dmexd 7334 | . . 3 ⊢ (𝐴 ∈ 𝑊 → dom (𝐼 × 2𝑜) ∈ V) |
10 | 3, 9 | syl5eqelr 2884 | . 2 ⊢ (𝐴 ∈ 𝑊 → 𝐼 ∈ V) |
11 | fvi 6481 | . . . 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 2846 | . 2 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2𝑜)) |
14 | 10, 13 | jca 508 | 1 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2972 Vcvv 3386 ∅c0 4116 I cid 5220 × cxp 5311 dom cdm 5313 ‘cfv 6102 2𝑜c2o 7794 Word cword 13533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-1st 7402 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-2o 7801 df-er 7983 df-map 8098 df-pm 8099 df-en 8197 df-dom 8198 df-sdom 8199 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-nn 11314 df-n0 11580 df-z 11666 df-uz 11930 df-fz 12580 df-fzo 12720 df-word 13534 |
This theorem is referenced by: efglem 18441 efgval 18442 efgtf 18447 efginvrel2 18452 efginvrel1 18453 efgredlemc 18471 efgcpbllemb 18482 efgcpbl2 18484 frgpcpbl 18486 frgpeccl 18488 frgpadd 18490 frgpinv 18491 frgpuplem 18499 frgpup1 18502 frgpnabllem1 18590 |
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