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Mirrors > Home > MPE Home > Th. List > wkslem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.) |
Ref | Expression |
---|---|
wkslem1 | ⊢ (𝐴 = 𝐵 → (if-((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)}, {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘(𝐵 + 1)), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹‘𝐵))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6907 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑃‘𝐴) = (𝑃‘𝐵)) | |
2 | fvoveq1 7454 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑃‘(𝐴 + 1)) = (𝑃‘(𝐵 + 1))) | |
3 | 1, 2 | eqeq12d 2751 | . 2 ⊢ (𝐴 = 𝐵 → ((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃‘𝐵) = (𝑃‘(𝐵 + 1)))) |
4 | 2fveq3 6912 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐼‘(𝐹‘𝐴)) = (𝐼‘(𝐹‘𝐵))) | |
5 | 1 | sneqd 4643 | . . 3 ⊢ (𝐴 = 𝐵 → {(𝑃‘𝐴)} = {(𝑃‘𝐵)}) |
6 | 4, 5 | eqeq12d 2751 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)} ↔ (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)})) |
7 | 1, 2 | preq12d 4746 | . . 3 ⊢ (𝐴 = 𝐵 → {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))}) |
8 | 7, 4 | sseq12d 4029 | . 2 ⊢ (𝐴 = 𝐵 → ({(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴)) ↔ {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹‘𝐵)))) |
9 | 3, 6, 8 | ifpbi123d 1078 | 1 ⊢ (𝐴 = 𝐵 → (if-((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)}, {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘(𝐵 + 1)), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹‘𝐵))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 if-wif 1062 = wceq 1537 ⊆ wss 3963 {csn 4631 {cpr 4633 ‘cfv 6563 (class class class)co 7431 1c1 11154 + caddc 11156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: wlk1walk 29672 wlkres 29703 wlkp1lem8 29713 crctcshwlkn0lem6 29845 crctcshwlkn0lem7 29846 crctcshwlkn0 29851 pfxwlk 35108 revwlk 35109 |
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