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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj965 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj852 34935. 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 |
|---|---|
| bnj965.1 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
| bnj965.2 | ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓) |
| bnj965.12000 | ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) |
| bnj965.13000 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) |
| Ref | Expression |
|---|---|
| bnj965 | ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj965.1 | . 2 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
| 2 | bnj965.2 | . 2 ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓) | |
| 3 | bnj965.13000 | . . 3 ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) | |
| 4 | 3 | bnj918 34780 | . 2 ⊢ 𝐺 ∈ V |
| 5 | bnj965.12000 | . 2 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) | |
| 6 | 1, 2, 4, 5, 3 | bnj1000 34955 | 1 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘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 ∪ cun 3949 {csn 4626 〈cop 4632 ∪ 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-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: bnj964 34957 bnj999 34972 |
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