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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj911 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 35002. 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 |
---|---|
bnj911.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
bnj911.2 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
Ref | Expression |
---|---|
bnj911 | ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∀𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj911.2 | . . 3 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
2 | 1 | bnj1095 34773 | . 2 ⊢ (𝜓 → ∀𝑖𝜓) |
3 | 2 | bnj1350 34817 | 1 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∀𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 ∀wal 1534 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ∅c0 4338 ∪ ciun 4995 suc csuc 6387 Fn wfn 6557 ‘cfv 6562 ωcom 7886 predc-bnj14 34680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-10 2138 ax-12 2174 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-ex 1776 df-nf 1780 df-ral 3059 |
This theorem is referenced by: bnj916 34925 bnj1014 34953 |
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