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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj126 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj150 33155. 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 3787 | . 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 33143 | . . 3 ⊢ 𝐹 ∈ V |
7 | 4, 6 | bnj106 33147 | . 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 296 | 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 1540 ∈ wcel 2105 ∀wral 3061 [wsbc 3727 ∅c0 4269 {csn 4573 ⟨cop 4579 ∪ ciun 4941 suc csuc 6304 ‘cfv 6479 ωcom 7780 1oc1o 8360 predc-bnj14 32967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ral 3062 df-rex 3071 df-v 3443 df-sbc 3728 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-pw 4549 df-sn 4574 df-pr 4576 df-uni 4853 df-iun 4943 df-br 5093 df-suc 6308 df-iota 6431 df-fv 6487 df-1o 8367 |
This theorem is referenced by: bnj150 33155 bnj153 33159 |
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