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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj557 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj852 34935. 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 |
|---|---|
| bnj557.3 | ⊢ 𝐷 = (ω ∖ {∅}) |
| bnj557.16 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)〉}) |
| bnj557.17 | ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)) |
| bnj557.18 | ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚)) |
| bnj557.19 | ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)) |
| bnj557.20 | ⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖)) |
| bnj557.21 | ⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) |
| bnj557.22 | ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) |
| bnj557.23 | ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) |
| bnj557.24 | ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅) |
| bnj557.25 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑚, 𝐶〉}) |
| bnj557.28 | ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
| bnj557.29 | ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
| bnj557.36 | ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛) |
| Ref | Expression |
|---|---|
| bnj557 | ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → (𝐺‘𝑚) = 𝐿) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3an4anass 1105 | . . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ 𝜁) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁))) | |
| 2 | bnj557.18 | . . . . . . . 8 ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚)) | |
| 3 | bnj557.19 | . . . . . . . 8 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)) | |
| 4 | 2, 3 | bnj556 34914 | . . . . . . 7 ⊢ (𝜂 → 𝜎) |
| 5 | 4 | 3anim3i 1155 | . . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎)) |
| 6 | bnj557.20 | . . . . . . 7 ⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖)) | |
| 7 | vex 3484 | . . . . . . . 8 ⊢ 𝑖 ∈ V | |
| 8 | 7 | bnj216 34746 | . . . . . . 7 ⊢ (𝑚 = suc 𝑖 → 𝑖 ∈ 𝑚) |
| 9 | 6, 8 | bnj837 34775 | . . . . . 6 ⊢ (𝜁 → 𝑖 ∈ 𝑚) |
| 10 | 5, 9 | anim12i 613 | . . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ 𝜁) → ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚)) |
| 11 | 1, 10 | sylbir 235 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁)) → ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚)) |
| 12 | 3 | bnj1254 34823 | . . . . . 6 ⊢ (𝜂 → 𝑚 = suc 𝑝) |
| 13 | 6 | simp3bi 1148 | . . . . . 6 ⊢ (𝜁 → 𝑚 = suc 𝑖) |
| 14 | bnj551 34756 | . . . . . 6 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → 𝑝 = 𝑖) | |
| 15 | 12, 13, 14 | syl2an 596 | . . . . 5 ⊢ ((𝜂 ∧ 𝜁) → 𝑝 = 𝑖) |
| 16 | 15 | adantl 481 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁)) → 𝑝 = 𝑖) |
| 17 | 11, 16 | jca 511 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁)) → (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) ∧ 𝑝 = 𝑖)) |
| 18 | bnj256 34720 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁))) | |
| 19 | df-3an 1089 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) ↔ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) ∧ 𝑝 = 𝑖)) | |
| 20 | 17, 18, 19 | 3imtr4i 292 | . 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖)) |
| 21 | bnj557.28 | . . 3 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | |
| 22 | bnj557.29 | . . 3 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
| 23 | bnj557.3 | . . 3 ⊢ 𝐷 = (ω ∖ {∅}) | |
| 24 | bnj557.16 | . . 3 ⊢ 𝐺 = (𝑓 ∪ {〈𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)〉}) | |
| 25 | bnj557.17 | . . 3 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)) | |
| 26 | bnj557.22 | . . 3 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) | |
| 27 | bnj557.25 | . . 3 ⊢ 𝐺 = (𝑓 ∪ {〈𝑚, 𝐶〉}) | |
| 28 | bnj557.21 | . . 3 ⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) | |
| 29 | bnj557.23 | . . 3 ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) | |
| 30 | bnj557.24 | . . 3 ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅) | |
| 31 | bnj557.36 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛) | |
| 32 | 21, 22, 23, 24, 25, 2, 26, 27, 28, 29, 30, 31 | bnj553 34912 | . 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → (𝐺‘𝑚) = 𝐿) |
| 33 | 20, 32 | syl 17 | 1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → (𝐺‘𝑚) = 𝐿) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∖ cdif 3948 ∪ cun 3949 ∅c0 4333 {csn 4626 〈cop 4632 ∪ ciun 4991 suc csuc 6386 Fn wfn 6556 ‘cfv 6561 ωcom 7887 ∧ w-bnj17 34700 predc-bnj14 34702 FrSe w-bnj15 34706 |
| 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-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-reg 9632 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-id 5578 df-eprel 5584 df-fr 5637 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-bnj17 34701 |
| This theorem is referenced by: bnj558 34916 |
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