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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj518 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj852 34461. 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 |
---|---|
bnj518.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
bnj518.2 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
bnj518.3 | ⊢ (𝜏 ↔ (𝜑 ∧ 𝜓 ∧ 𝑛 ∈ ω ∧ 𝑝 ∈ 𝑛)) |
Ref | Expression |
---|---|
bnj518 | ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj518.3 | . . . 4 ⊢ (𝜏 ↔ (𝜑 ∧ 𝜓 ∧ 𝑛 ∈ ω ∧ 𝑝 ∈ 𝑛)) | |
2 | bnj334 34253 | . . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑛 ∈ ω ∧ 𝑝 ∈ 𝑛) ↔ (𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛)) | |
3 | 1, 2 | bitri 275 | . . 3 ⊢ (𝜏 ↔ (𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛)) |
4 | df-bnj17 34227 | . . . 4 ⊢ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛) ↔ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑝 ∈ 𝑛)) | |
5 | bnj518.1 | . . . . . 6 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | |
6 | bnj518.2 | . . . . . 6 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
7 | 5, 6 | bnj517 34425 | . . . . 5 ⊢ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓) → ∀𝑝 ∈ 𝑛 (𝑓‘𝑝) ⊆ 𝐴) |
8 | 7 | r19.21bi 3242 | . . . 4 ⊢ (((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑝 ∈ 𝑛) → (𝑓‘𝑝) ⊆ 𝐴) |
9 | 4, 8 | sylbi 216 | . . 3 ⊢ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛) → (𝑓‘𝑝) ⊆ 𝐴) |
10 | 3, 9 | sylbi 216 | . 2 ⊢ (𝜏 → (𝑓‘𝑝) ⊆ 𝐴) |
11 | ssel 3970 | . . . 4 ⊢ ((𝑓‘𝑝) ⊆ 𝐴 → (𝑦 ∈ (𝑓‘𝑝) → 𝑦 ∈ 𝐴)) | |
12 | bnj93 34403 | . . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑦 ∈ 𝐴) → pred(𝑦, 𝐴, 𝑅) ∈ V) | |
13 | 12 | ex 412 | . . . 4 ⊢ (𝑅 FrSe 𝐴 → (𝑦 ∈ 𝐴 → pred(𝑦, 𝐴, 𝑅) ∈ V)) |
14 | 11, 13 | sylan9r 508 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝑓‘𝑝) ⊆ 𝐴) → (𝑦 ∈ (𝑓‘𝑝) → pred(𝑦, 𝐴, 𝑅) ∈ V)) |
15 | 14 | ralrimiv 3139 | . 2 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝑓‘𝑝) ⊆ 𝐴) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) |
16 | 10, 15 | sylan2 592 | 1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 Vcvv 3468 ⊆ wss 3943 ∅c0 4317 ∪ ciun 4990 suc csuc 6359 ‘cfv 6536 ωcom 7851 ∧ w-bnj17 34226 predc-bnj14 34228 FrSe w-bnj15 34232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-tr 5259 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fv 6544 df-om 7852 df-bnj17 34227 df-bnj14 34229 df-bnj13 34231 df-bnj15 34233 |
This theorem is referenced by: bnj535 34430 bnj546 34436 |
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