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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1112 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32392. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1112.1 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
Ref | Expression |
---|---|
bnj1112 | ⊢ (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1112.1 | . . 3 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
2 | 1 | bnj115 32105 | . 2 ⊢ (𝜓 ↔ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
3 | eleq1w 2872 | . . . . 5 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ ω ↔ 𝑗 ∈ ω)) | |
4 | suceq 6224 | . . . . . 6 ⊢ (𝑖 = 𝑗 → suc 𝑖 = suc 𝑗) | |
5 | 4 | eleq1d 2874 | . . . . 5 ⊢ (𝑖 = 𝑗 → (suc 𝑖 ∈ 𝑛 ↔ suc 𝑗 ∈ 𝑛)) |
6 | 3, 5 | anbi12d 633 | . . . 4 ⊢ (𝑖 = 𝑗 → ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ↔ (𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛))) |
7 | 4 | fveq2d 6649 | . . . . 5 ⊢ (𝑖 = 𝑗 → (𝑓‘suc 𝑖) = (𝑓‘suc 𝑗)) |
8 | fveq2 6645 | . . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑓‘𝑖) = (𝑓‘𝑗)) | |
9 | 8 | bnj1113 32167 | . . . . 5 ⊢ (𝑖 = 𝑗 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)) |
10 | 7, 9 | eqeq12d 2814 | . . . 4 ⊢ (𝑖 = 𝑗 → ((𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))) |
11 | 6, 10 | imbi12d 348 | . . 3 ⊢ (𝑖 = 𝑗 → (((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))) |
12 | 11 | cbvalvw 2043 | . 2 ⊢ (∀𝑖((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))) |
13 | 2, 12 | bitri 278 | 1 ⊢ (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∪ ciun 4881 suc csuc 6161 ‘cfv 6324 ωcom 7560 predc-bnj14 32068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-suc 6165 df-iota 6283 df-fv 6332 |
This theorem is referenced by: bnj1118 32366 |
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