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Mirrors > Home > MPE Home > Th. List > 2wlklem | Structured version Visualization version GIF version |
Description: Lemma for theorems for walks of length 2. (Contributed by Alexander van der Vekens, 1-Feb-2018.) |
Ref | Expression |
---|---|
2wlklem | ⊢ (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 10824 | . 2 ⊢ 0 ∈ V | |
2 | 1ex 10826 | . 2 ⊢ 1 ∈ V | |
3 | 2fveq3 6719 | . . 3 ⊢ (𝑘 = 0 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘0))) | |
4 | fveq2 6714 | . . . 4 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0)) | |
5 | fv0p1e1 11950 | . . . 4 ⊢ (𝑘 = 0 → (𝑃‘(𝑘 + 1)) = (𝑃‘1)) | |
6 | 4, 5 | preq12d 4654 | . . 3 ⊢ (𝑘 = 0 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘0), (𝑃‘1)}) |
7 | 3, 6 | eqeq12d 2753 | . 2 ⊢ (𝑘 = 0 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)})) |
8 | 2fveq3 6719 | . . 3 ⊢ (𝑘 = 1 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘1))) | |
9 | fveq2 6714 | . . . 4 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1)) | |
10 | oveq1 7217 | . . . . . 6 ⊢ (𝑘 = 1 → (𝑘 + 1) = (1 + 1)) | |
11 | 1p1e2 11952 | . . . . . 6 ⊢ (1 + 1) = 2 | |
12 | 10, 11 | eqtrdi 2794 | . . . . 5 ⊢ (𝑘 = 1 → (𝑘 + 1) = 2) |
13 | 12 | fveq2d 6718 | . . . 4 ⊢ (𝑘 = 1 → (𝑃‘(𝑘 + 1)) = (𝑃‘2)) |
14 | 9, 13 | preq12d 4654 | . . 3 ⊢ (𝑘 = 1 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘1), (𝑃‘2)}) |
15 | 8, 14 | eqeq12d 2753 | . 2 ⊢ (𝑘 = 1 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) |
16 | 1, 2, 7, 15 | ralpr 4613 | 1 ⊢ (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∀wral 3058 {cpr 4540 ‘cfv 6377 (class class class)co 7210 0cc0 10726 1c1 10727 + caddc 10729 2c2 11882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 df-po 5465 df-so 5466 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-ov 7213 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-pnf 10866 df-mnf 10867 df-ltxr 10869 df-2 11890 |
This theorem is referenced by: upgr2wlk 27753 usgr2wlkneq 27840 usgr2trlncl 27844 usgr2pthlem 27847 usgr2pth 27848 uspgrn2crct 27889 wlk2v2elem2 28236 |
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