Theorem wkslem2 16245

Description: Lemma 2 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.)

Assertion

Ref	Expression
wkslem2	⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (if-((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)}, {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘𝐶), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘𝐶)} ⊆ (𝐼‘(𝐹‘𝐵)))))

Proof of Theorem wkslem2

Step	Hyp	Ref	Expression
1		fveq2 5648	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑃‘𝐴) = (𝑃‘𝐵))
2	1	adantr 276	. . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃‘𝐴) = (𝑃‘𝐵))
3		fveq2 5648	. . . 4 ⊢ ((𝐴 + 1) = 𝐶 → (𝑃‘(𝐴 + 1)) = (𝑃‘𝐶))
4	3	adantl 277	. . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃‘(𝐴 + 1)) = (𝑃‘𝐶))
5	2, 4	eqeq12d 2246	. 2 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃‘𝐵) = (𝑃‘𝐶)))
6		2fveq3 5653	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐼‘(𝐹‘𝐴)) = (𝐼‘(𝐹‘𝐵)))
7	1	sneqd 3686	. . . 4 ⊢ (𝐴 = 𝐵 → {(𝑃‘𝐴)} = {(𝑃‘𝐵)})
8	6, 7	eqeq12d 2246	. . 3 ⊢ (𝐴 = 𝐵 → ((𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)} ↔ (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}))
9	8	adantr 276	. 2 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)} ↔ (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}))
10	2, 4	preq12d 3760	. . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃‘𝐵), (𝑃‘𝐶)})
11	6	adantr 276	. . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝐼‘(𝐹‘𝐴)) = (𝐼‘(𝐹‘𝐵)))
12	10, 11	sseq12d 3259	. 2 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ({(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴)) ↔ {(𝑃‘𝐵), (𝑃‘𝐶)} ⊆ (𝐼‘(𝐹‘𝐵))))
13	5, 9, 12	ifpbi123d 1001	1 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (if-((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)}, {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘𝐶), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘𝐶)} ⊆ (𝐼‘(𝐹‘𝐵)))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  if-wif 986   = wceq 1398   ⊆ wss 3201  {csn 3673  {cpr 3674  ‘cfv 5333  (class class class)co 6028  1c1 8076   + caddc 8078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-ifp 987  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341

This theorem is referenced by:  wlkl1loop  16282  wlk1walkdom  16283