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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj126 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj150 33918. 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 3838 | . 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 33906 | . . 3 ⊢ 𝐹 ∈ V |
7 | 4, 6 | bnj106 33910 | . 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 205 = wceq 1542 ∈ wcel 2107 ∀wral 3062 [wsbc 3778 ∅c0 4323 {csn 4629 ⟨cop 4635 ∪ ciun 4998 suc csuc 6367 ‘cfv 6544 ωcom 7855 1oc1o 8459 predc-bnj14 33730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-pw 4605 df-sn 4630 df-pr 4632 df-uni 4910 df-iun 5000 df-br 5150 df-suc 6371 df-iota 6496 df-fv 6552 df-1o 8466 |
This theorem is referenced by: bnj150 33918 bnj153 33922 |
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