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Theorem wwlknonOLD 26990
 Description: Obsolete version of iswwlksnon 26982 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
wwlknon.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
wwlknonOLD ((𝐴𝑉𝐵𝑉) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵)))

Proof of Theorem wwlknonOLD
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 wwlknon.v . . . 4 𝑉 = (Vtx‘𝐺)
21iswwlksnonOLD 26983 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
32eleq2d 2836 . 2 ((𝐴𝑉𝐵𝑉) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ↔ 𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)}))
4 fveq1 6331 . . . . . 6 (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
54eqeq1d 2773 . . . . 5 (𝑤 = 𝑊 → ((𝑤‘0) = 𝐴 ↔ (𝑊‘0) = 𝐴))
6 fveq1 6331 . . . . . 6 (𝑤 = 𝑊 → (𝑤𝑁) = (𝑊𝑁))
76eqeq1d 2773 . . . . 5 (𝑤 = 𝑊 → ((𝑤𝑁) = 𝐵 ↔ (𝑊𝑁) = 𝐵))
85, 7anbi12d 608 . . . 4 (𝑤 = 𝑊 → (((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵) ↔ ((𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵)))
98elrab 3515 . . 3 (𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵)))
10 3anass 1080 . . 3 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵)))
119, 10bitr4i 267 . 2 (𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵))
123, 11syl6bb 276 1 ((𝐴𝑉𝐵𝑉) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  {crab 3065  ‘cfv 6031  (class class class)co 6793  0cc0 10138  Vtxcvtx 26095   WWalksN cwwlksn 26954   WWalksNOn cwwlksnon 26955 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-wwlksn 26959  df-wwlksnon 26960 This theorem is referenced by:  wwlksnwwlksnonOLD  27062  wspthsnwspthsnonOLD  27063  elwwlks2ons3OLD  27103  wpthswwlks2onOLD  27110
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