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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1039 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj69 35024. 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 |
|---|---|
| bnj1039.1 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
| bnj1039.2 | ⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓) |
| Ref | Expression |
|---|---|
| bnj1039 | ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj1039.2 | . 2 ⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓) | |
| 2 | vex 3484 | . . 3 ⊢ 𝑗 ∈ V | |
| 3 | bnj1039.1 | . . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
| 4 | nfra1 3284 | . . . 4 ⊢ Ⅎ𝑖∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) | |
| 5 | 3, 4 | nfxfr 1853 | . . 3 ⊢ Ⅎ𝑖𝜓 |
| 6 | 2, 5 | sbcgfi 3864 | . 2 ⊢ ([𝑗 / 𝑖]𝜓 ↔ 𝜓) |
| 7 | 1, 6, 3 | 3bitri 297 | 1 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∀wral 3061 [wsbc 3788 ∪ ciun 4991 suc csuc 6386 ‘cfv 6561 ωcom 7887 predc-bnj14 34702 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-sbc 3789 |
| This theorem is referenced by: bnj1128 35004 |
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