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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1121 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj69 35024. 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 |
|---|---|
| bnj1121.1 | ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)) |
| bnj1121.2 | ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)) |
| bnj1121.3 | ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
| bnj1121.4 | ⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) |
| bnj1121.5 | ⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵)) |
| bnj1121.6 | ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 𝜂) |
| bnj1121.7 | ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
| Ref | Expression |
|---|---|
| bnj1121 | ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 19.8a 2181 | . . . . 5 ⊢ (𝜒 → ∃𝑛𝜒) | |
| 2 | 1 | bnj707 34769 | . . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∃𝑛𝜒) |
| 3 | bnj1121.3 | . . . . 5 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
| 4 | bnj1121.7 | . . . . 5 ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
| 5 | 3, 4 | bnj1083 34992 | . . . 4 ⊢ (𝑓 ∈ 𝐾 ↔ ∃𝑛𝜒) |
| 6 | 2, 5 | sylibr 234 | . . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑓 ∈ 𝐾) |
| 7 | bnj1121.4 | . . . . . 6 ⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) | |
| 8 | 7 | simplbi 497 | . . . . 5 ⊢ (𝜁 → 𝑖 ∈ 𝑛) |
| 9 | 8 | bnj708 34770 | . . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑖 ∈ 𝑛) |
| 10 | 3 | bnj1235 34818 | . . . . . 6 ⊢ (𝜒 → 𝑓 Fn 𝑛) |
| 11 | 10 | bnj707 34769 | . . . . 5 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑓 Fn 𝑛) |
| 12 | 11 | fndmd 6673 | . . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → dom 𝑓 = 𝑛) |
| 13 | 9, 12 | eleqtrrd 2844 | . . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑖 ∈ dom 𝑓) |
| 14 | bnj1121.6 | . . . . 5 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 𝜂) | |
| 15 | 14, 9 | bnj1294 34831 | . . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝜂) |
| 16 | bnj1121.5 | . . . 4 ⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵)) | |
| 17 | 15, 16 | sylib 218 | . . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵)) |
| 18 | 6, 13, 17 | mp2and 699 | . 2 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → (𝑓‘𝑖) ⊆ 𝐵) |
| 19 | 7 | simprbi 496 | . . 3 ⊢ (𝜁 → 𝑧 ∈ (𝑓‘𝑖)) |
| 20 | 19 | bnj708 34770 | . 2 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ (𝑓‘𝑖)) |
| 21 | 18, 20 | sseldd 3984 | 1 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 dom cdm 5685 Fn wfn 6556 ‘cfv 6561 ∧ w-bnj17 34700 predc-bnj14 34702 FrSe w-bnj15 34706 TrFow-bnj19 34710 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-ss 3968 df-fn 6564 df-bnj17 34701 |
| This theorem is referenced by: bnj1030 35001 |
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