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Theorem iswwlksnonOLD 26879
Description: Obsolete version of iswwlksnon 26878 as of 14-Mar-2022. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
iswwlksnon.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
iswwlksnonOLD ((𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
Distinct variable groups:   𝑤,𝐴   𝑤,𝐵   𝑤,𝐺   𝑤,𝑁   𝑤,𝑉

Proof of Theorem iswwlksnonOLD
Dummy variables 𝑎 𝑏 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iswwlksnon.v . . . . . 6 𝑉 = (Vtx‘𝐺)
21wwlksnon 26876 . . . . 5 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))
32adantr 472 . . . 4 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝑁 WWalksNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))
4 eqeq2 2735 . . . . . . 7 (𝑎 = 𝐴 → ((𝑤‘0) = 𝑎 ↔ (𝑤‘0) = 𝐴))
5 eqeq2 2735 . . . . . . 7 (𝑏 = 𝐵 → ((𝑤𝑁) = 𝑏 ↔ (𝑤𝑁) = 𝐵))
64, 5bi2anan9 953 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)))
76rabbidv 3293 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
87adantl 473 . . . 4 ((((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
9 simprl 811 . . . 4 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → 𝐴𝑉)
10 simprr 813 . . . 4 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → 𝐵𝑉)
11 ovex 6793 . . . . . 6 (𝑁 WWalksN 𝐺) ∈ V
1211rabex 4920 . . . . 5 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} ∈ V
1312a1i 11 . . . 4 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} ∈ V)
143, 8, 9, 10, 13ovmpt2d 6905 . . 3 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
1514ex 449 . 2 ((𝑁 ∈ ℕ0𝐺 ∈ V) → ((𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)}))
16 0ov 6797 . . . 4 (𝐴𝐵) = ∅
17 df-wwlksnon 26856 . . . . . 6 WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}))
1817mpt2ndm0 6992 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksNOn 𝐺) = ∅)
1918oveqd 6782 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = (𝐴𝐵))
20 df-wwlksn 26855 . . . . . . 7 WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
2120mpt2ndm0 6992 . . . . . 6 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = ∅)
2221rabeqdv 3298 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
23 rab0 4063 . . . . 5 {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = ∅
2422, 23syl6eq 2774 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = ∅)
2516, 19, 243eqtr4a 2784 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
2625a1d 25 . 2 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → ((𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)}))
2715, 26pm2.61i 176 1 ((𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1596  wcel 2103  {crab 3018  Vcvv 3304  c0 4023  cfv 6001  (class class class)co 6765  cmpt2 6767  0cc0 10049  1c1 10050   + caddc 10052  0cn0 11405  chash 13232  Vtxcvtx 25994  WWalkscwwlks 26849   WWalksN cwwlksn 26850   WWalksNOn cwwlksnon 26851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-wwlksn 26855  df-wwlksnon 26856
This theorem is referenced by:  wwlknonOLD  26886  wpthswwlks2onOLD  27004
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