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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj126 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj150 30200. 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 3458 | . 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 30188 | . . 3 ⊢ 𝐹 ∈ V |
7 | 4, 6 | bnj106 30192 | . 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 285 | 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 195 = wceq 1475 ∈ wcel 1977 ∀wral 2896 [wsbc 3402 ∅c0 3874 {csn 4125 〈cop 4131 ∪ ciun 4455 suc csuc 5642 ‘cfv 5804 ωcom 6957 1𝑜c1o 7440 predc-bnj14 30007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-pw 4110 df-sn 4126 df-pr 4128 df-uni 4373 df-iun 4457 df-br 4584 df-suc 5646 df-iota 5768 df-fv 5812 df-1o 7447 |
This theorem is referenced by: bnj150 30200 bnj153 30204 |
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