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Mirrors > Home > MPE Home > Th. List > efgrcl | Structured version Visualization version GIF version |
Description: Lemma for efgval 17953. (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 7456 | . . . 4 ⊢ 2𝑜 ≠ ∅ | |
2 | dmxp 5265 | . . . 4 ⊢ (2𝑜 ≠ ∅ → dom (𝐼 × 2𝑜) = 𝐼) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ dom (𝐼 × 2𝑜) = 𝐼 |
4 | elfvex 6131 | . . . . . 6 ⊢ (𝐴 ∈ ( I ‘Word (𝐼 × 2𝑜)) → Word (𝐼 × 2𝑜) ∈ V) | |
5 | efgval.w | . . . . . 6 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜)) | |
6 | 4, 5 | eleq2s 2706 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → Word (𝐼 × 2𝑜) ∈ V) |
7 | wrdexb 13171 | . . . . 5 ⊢ ((𝐼 × 2𝑜) ∈ V ↔ Word (𝐼 × 2𝑜) ∈ V) | |
8 | 6, 7 | sylibr 223 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐼 × 2𝑜) ∈ V) |
9 | dmexg 6989 | . . . 4 ⊢ ((𝐼 × 2𝑜) ∈ V → dom (𝐼 × 2𝑜) ∈ V) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑊 → dom (𝐼 × 2𝑜) ∈ V) |
11 | 3, 10 | syl5eqelr 2693 | . 2 ⊢ (𝐴 ∈ 𝑊 → 𝐼 ∈ V) |
12 | fvi 6165 | . . . 4 ⊢ (Word (𝐼 × 2𝑜) ∈ V → ( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜)) | |
13 | 6, 12 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑊 → ( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜)) |
14 | 5, 13 | syl5eq 2656 | . 2 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2𝑜)) |
15 | 11, 14 | jca 553 | 1 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∅c0 3874 I cid 4948 × cxp 5036 dom cdm 5038 ‘cfv 5804 2𝑜c2o 7441 Word cword 13146 |
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-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
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-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-2o 7448 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-word 13154 |
This theorem is referenced by: efglem 17952 efgval 17953 efgtf 17958 efginvrel2 17963 efginvrel1 17964 efgredlemc 17981 efgcpbllemb 17991 efgcpbl2 17993 frgpcpbl 17995 frgpeccl 17997 frgpadd 17999 frgpinv 18000 frgpuplem 18008 frgpup1 18011 frgpnabllem1 18099 |
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