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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj590 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj852 30245. 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 |
---|---|
bnj590.1 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
Ref | Expression |
---|---|
bnj590 | ⊢ ((𝐵 = suc 𝑖 ∧ 𝜓) → (𝑖 ∈ ω → (𝐵 ∈ 𝑛 → (𝑓‘𝐵) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj590.1 | . . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
2 | rsp 2913 | . . . 4 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑖 ∈ ω → (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))) | |
3 | 1, 2 | sylbi 206 | . . 3 ⊢ (𝜓 → (𝑖 ∈ ω → (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))) |
4 | eleq1 2676 | . . . . 5 ⊢ (𝐵 = suc 𝑖 → (𝐵 ∈ 𝑛 ↔ suc 𝑖 ∈ 𝑛)) | |
5 | fveq2 6103 | . . . . . 6 ⊢ (𝐵 = suc 𝑖 → (𝑓‘𝐵) = (𝑓‘suc 𝑖)) | |
6 | 5 | eqeq1d 2612 | . . . . 5 ⊢ (𝐵 = suc 𝑖 → ((𝑓‘𝐵) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
7 | 4, 6 | imbi12d 333 | . . . 4 ⊢ (𝐵 = suc 𝑖 → ((𝐵 ∈ 𝑛 → (𝑓‘𝐵) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))) |
8 | 7 | imbi2d 329 | . . 3 ⊢ (𝐵 = suc 𝑖 → ((𝑖 ∈ ω → (𝐵 ∈ 𝑛 → (𝑓‘𝐵) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑖 ∈ ω → (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))) |
9 | 3, 8 | syl5ibr 235 | . 2 ⊢ (𝐵 = suc 𝑖 → (𝜓 → (𝑖 ∈ ω → (𝐵 ∈ 𝑛 → (𝑓‘𝐵) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))) |
10 | 9 | imp 444 | 1 ⊢ ((𝐵 = suc 𝑖 ∧ 𝜓) → (𝑖 ∈ ω → (𝐵 ∈ 𝑛 → (𝑓‘𝐵) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∪ ciun 4455 suc csuc 5642 ‘cfv 5804 ωcom 6957 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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
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-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 |
This theorem is referenced by: bnj594 30236 |
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