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Mirrors > Home > MPE Home > Th. List > opfi1uzind | Structured version Visualization version GIF version |
Description: Properties of an ordered pair with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as orderd pairs of vertices and edges) with a finite number of vertices, usually with 𝐿 = 0 (see opfi1ind 13139) or 𝐿 = 1. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.) |
Ref | Expression |
---|---|
opfi1uzind.e | ⊢ 𝐸 ∈ V |
opfi1uzind.f | ⊢ 𝐹 ∈ V |
opfi1uzind.l | ⊢ 𝐿 ∈ ℕ0 |
opfi1uzind.1 | ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑)) |
opfi1uzind.2 | ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃)) |
opfi1uzind.3 | ⊢ ((〈𝑣, 𝑒〉 ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺) |
opfi1uzind.4 | ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒)) |
opfi1uzind.base | ⊢ ((〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (#‘𝑣) = 𝐿) → 𝜓) |
opfi1uzind.step | ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓) |
Ref | Expression |
---|---|
opfi1uzind | ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . 5 ⊢ (𝑉 ∈ Fin → 𝑉 ∈ Fin) | |
2 | opfi1uzind.e | . . . . . . . 8 ⊢ 𝐸 ∈ V | |
3 | 2 | a1i 11 | . . . . . . 7 ⊢ (𝑎 = 𝑉 → 𝐸 ∈ V) |
4 | opeq12 4342 | . . . . . . . 8 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → 〈𝑎, 𝑏〉 = 〈𝑉, 𝐸〉) | |
5 | 4 | eleq1d 2672 | . . . . . . 7 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → (〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑉, 𝐸〉 ∈ 𝐺)) |
6 | 3, 5 | sbcied 3439 | . . . . . 6 ⊢ (𝑎 = 𝑉 → ([𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑉, 𝐸〉 ∈ 𝐺)) |
7 | 6 | adantl 481 | . . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝑎 = 𝑉) → ([𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑉, 𝐸〉 ∈ 𝐺)) |
8 | 1, 7 | sbcied 3439 | . . . 4 ⊢ (𝑉 ∈ Fin → ([𝑉 / 𝑎][𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑉, 𝐸〉 ∈ 𝐺)) |
9 | 8 | biimparc 503 | . . 3 ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin) → [𝑉 / 𝑎][𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺) |
10 | 9 | 3adant3 1074 | . 2 ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → [𝑉 / 𝑎][𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺) |
11 | opfi1uzind.f | . . 3 ⊢ 𝐹 ∈ V | |
12 | opfi1uzind.l | . . 3 ⊢ 𝐿 ∈ ℕ0 | |
13 | opfi1uzind.1 | . . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑)) | |
14 | opfi1uzind.2 | . . 3 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃)) | |
15 | vex 3176 | . . . . . 6 ⊢ 𝑣 ∈ V | |
16 | vex 3176 | . . . . . 6 ⊢ 𝑒 ∈ V | |
17 | opeq12 4342 | . . . . . . 7 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → 〈𝑎, 𝑏〉 = 〈𝑣, 𝑒〉) | |
18 | 17 | eleq1d 2672 | . . . . . 6 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → (〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑣, 𝑒〉 ∈ 𝐺)) |
19 | 15, 16, 18 | sbc2ie 3472 | . . . . 5 ⊢ ([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑣, 𝑒〉 ∈ 𝐺) |
20 | opfi1uzind.3 | . . . . 5 ⊢ ((〈𝑣, 𝑒〉 ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺) | |
21 | 19, 20 | sylanb 488 | . . . 4 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺) |
22 | difexg 4735 | . . . . . 6 ⊢ (𝑣 ∈ V → (𝑣 ∖ {𝑛}) ∈ V) | |
23 | 15, 22 | ax-mp 5 | . . . . 5 ⊢ (𝑣 ∖ {𝑛}) ∈ V |
24 | 11 | elexi 3186 | . . . . 5 ⊢ 𝐹 ∈ V |
25 | opeq12 4342 | . . . . . 6 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → 〈𝑎, 𝑏〉 = 〈(𝑣 ∖ {𝑛}), 𝐹〉) | |
26 | 25 | eleq1d 2672 | . . . . 5 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺)) |
27 | 23, 24, 26 | sbc2ie 3472 | . . . 4 ⊢ ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺) |
28 | 21, 27 | sylibr 223 | . . 3 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺) |
29 | opfi1uzind.4 | . . . 4 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒)) | |
30 | 29 | idi 2 | . . 3 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒)) |
31 | opfi1uzind.base | . . . 4 ⊢ ((〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (#‘𝑣) = 𝐿) → 𝜓) | |
32 | 19, 31 | sylanb 488 | . . 3 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ (#‘𝑣) = 𝐿) → 𝜓) |
33 | 19 | 3anbi1i 1246 | . . . . 5 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣) ↔ (〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) |
34 | 33 | anbi2i 726 | . . . 4 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ↔ ((𝑦 + 1) ∈ ℕ0 ∧ (〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))) |
35 | opfi1uzind.step | . . . 4 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓) | |
36 | 34, 35 | sylanb 488 | . . 3 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓) |
37 | 11, 12, 13, 14, 28, 30, 32, 36 | fi1uzind 13134 | . 2 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑) |
38 | 10, 37 | syld3an1 1364 | 1 ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 [wsbc 3402 ∖ cdif 3537 {csn 4125 〈cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 1c1 9816 + caddc 9818 ≤ cle 9954 ℕ0cn0 11169 #chash 12979 |
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-rmo 2904 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-int 4411 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-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 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-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 |
This theorem is referenced by: opfi1ind 13139 |
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