| Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj126 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj150 34890. 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 |
|---|---|
| bnj126.1 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
| bnj126.2 | ⊢ (𝜓′ ↔ [1o / 𝑛]𝜓) |
| bnj126.3 | ⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′) |
| bnj126.4 | ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} |
| Ref | Expression |
|---|---|
| bnj126 | ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj126.3 | . 2 ⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′) | |
| 2 | bnj126.2 | . . 3 ⊢ (𝜓′ ↔ [1o / 𝑛]𝜓) | |
| 3 | 2 | sbcbii 3846 | . 2 ⊢ ([𝐹 / 𝑓]𝜓′ ↔ [𝐹 / 𝑓][1o / 𝑛]𝜓) |
| 4 | bnj126.1 | . . 3 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
| 5 | bnj126.4 | . . . 4 ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} | |
| 6 | 5 | bnj95 34878 | . . 3 ⊢ 𝐹 ∈ V |
| 7 | 4, 6 | bnj106 34882 | . 2 ⊢ ([𝐹 / 𝑓][1o / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
| 8 | 1, 3, 7 | 3bitri 297 | 1 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (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 ∅c0 4333 {csn 4626 〈cop 4632 ∪ ciun 4991 suc csuc 6386 ‘cfv 6561 ωcom 7887 1oc1o 8499 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-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-pr 4629 df-uni 4908 df-iun 4993 df-br 5144 df-suc 6390 df-iota 6514 df-fv 6569 df-1o 8506 |
| This theorem is referenced by: bnj150 34890 bnj153 34894 |
| Copyright terms: Public domain | W3C validator |