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Theorem iswwlksnonOLD 26950
Description: Obsolete version of iswwlksnon 26949 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 26947 . . . . 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 2763 . . . . . . 7 (𝑎 = 𝐴 → ((𝑤‘0) = 𝑎 ↔ (𝑤‘0) = 𝐴))
5 eqeq2 2763 . . . . . . 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 3321 . . . . 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 6833 . . . . . 6 (𝑁 WWalksN 𝐺) ∈ V
1211rabex 4956 . . . . 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 6945 . . 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 6837 . . . 4 (𝐴𝐵) = ∅
17 df-wwlksnon 26927 . . . . . 6 WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}))
1817mpt2ndm0 7032 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksNOn 𝐺) = ∅)
1918oveqd 6822 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = (𝐴𝐵))
20 df-wwlksn 26926 . . . . . . 7 WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
2120mpt2ndm0 7032 . . . . . 6 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = ∅)
2221rabeqdv 3326 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
23 rab0 4090 . . . . 5 {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = ∅
2422, 23syl6eq 2802 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = ∅)
2516, 19, 243eqtr4a 2812 . . 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 1624  wcel 2131  {crab 3046  Vcvv 3332  c0 4050  cfv 6041  (class class class)co 6805  cmpt2 6807  0cc0 10120  1c1 10121   + caddc 10123  0cn0 11476  chash 13303  Vtxcvtx 26065  WWalkscwwlks 26920   WWalksN cwwlksn 26921   WWalksNOn cwwlksnon 26922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-wwlksn 26926  df-wwlksnon 26927
This theorem is referenced by:  wwlknonOLD  26957  wpthswwlks2onOLD  27075
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