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Mirrors > Home > MPE Home > Th. List > wkslem2 | Structured version Visualization version GIF version |
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.) |
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))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6825 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑃‘𝐴) = (𝑃‘𝐵)) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃‘𝐴) = (𝑃‘𝐵)) |
3 | fveq2 6825 | . . . 4 ⊢ ((𝐴 + 1) = 𝐶 → (𝑃‘(𝐴 + 1)) = (𝑃‘𝐶)) | |
4 | 3 | adantl 482 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃‘(𝐴 + 1)) = (𝑃‘𝐶)) |
5 | 2, 4 | eqeq12d 2752 | . 2 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃‘𝐵) = (𝑃‘𝐶))) |
6 | 2fveq3 6830 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐼‘(𝐹‘𝐴)) = (𝐼‘(𝐹‘𝐵))) | |
7 | 1 | sneqd 4585 | . . . 4 ⊢ (𝐴 = 𝐵 → {(𝑃‘𝐴)} = {(𝑃‘𝐵)}) |
8 | 6, 7 | eqeq12d 2752 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)} ↔ (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)})) |
9 | 8 | adantr 481 | . 2 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)} ↔ (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)})) |
10 | 2, 4 | preq12d 4689 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃‘𝐵), (𝑃‘𝐶)}) |
11 | 6 | adantr 481 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝐼‘(𝐹‘𝐴)) = (𝐼‘(𝐹‘𝐵))) |
12 | 10, 11 | sseq12d 3965 | . 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 1077 | 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))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 if-wif 1060 = wceq 1540 ⊆ wss 3898 {csn 4573 {cpr 4575 ‘cfv 6479 (class class class)co 7337 1c1 10973 + caddc 10975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-iota 6431 df-fv 6487 |
This theorem is referenced by: wlkl1loop 28294 wlk1walk 28295 crctcshwlkn0lem6 28468 1wlkdlem4 28792 |
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