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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj553 | 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 |
|---|---|
| bnj553.1 | ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
| bnj553.2 | ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
| bnj553.3 | ⊢ 𝐷 = (ω ∖ {∅}) |
| bnj553.4 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)〉}) |
| bnj553.5 | ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)) |
| bnj553.6 | ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚)) |
| bnj553.7 | ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) |
| bnj553.8 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑚, 𝐶〉}) |
| bnj553.9 | ⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) |
| bnj553.10 | ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) |
| bnj553.11 | ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅) |
| bnj553.12 | ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛) |
| Ref | Expression |
|---|---|
| bnj553 | ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → (𝐺‘𝑚) = 𝐿) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj553.12 | . . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛) | |
| 2 | 1 | fnfund 6669 | . . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → Fun 𝐺) |
| 3 | opex 5469 | . . . . . . 7 ⊢ 〈𝑚, 𝐶〉 ∈ V | |
| 4 | 3 | snid 4662 | . . . . . 6 ⊢ 〈𝑚, 𝐶〉 ∈ {〈𝑚, 𝐶〉} |
| 5 | elun2 4183 | . . . . . 6 ⊢ (〈𝑚, 𝐶〉 ∈ {〈𝑚, 𝐶〉} → 〈𝑚, 𝐶〉 ∈ (𝑓 ∪ {〈𝑚, 𝐶〉})) | |
| 6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ 〈𝑚, 𝐶〉 ∈ (𝑓 ∪ {〈𝑚, 𝐶〉}) |
| 7 | bnj553.8 | . . . . 5 ⊢ 𝐺 = (𝑓 ∪ {〈𝑚, 𝐶〉}) | |
| 8 | 6, 7 | eleqtrri 2840 | . . . 4 ⊢ 〈𝑚, 𝐶〉 ∈ 𝐺 |
| 9 | funopfv 6958 | . . . 4 ⊢ (Fun 𝐺 → (〈𝑚, 𝐶〉 ∈ 𝐺 → (𝐺‘𝑚) = 𝐶)) | |
| 10 | 2, 8, 9 | mpisyl 21 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝐺‘𝑚) = 𝐶) |
| 11 | 10 | 3ad2ant1 1134 | . 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → (𝐺‘𝑚) = 𝐶) |
| 12 | fveq2 6906 | . . . . . 6 ⊢ (𝑝 = 𝑖 → (𝐺‘𝑝) = (𝐺‘𝑖)) | |
| 13 | 12 | bnj1113 34799 | . . . . 5 ⊢ (𝑝 = 𝑖 → ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
| 14 | bnj553.11 | . . . . 5 ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅) | |
| 15 | bnj553.10 | . . . . 5 ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) | |
| 16 | 13, 14, 15 | 3eqtr4g 2802 | . . . 4 ⊢ (𝑝 = 𝑖 → 𝐿 = 𝐾) |
| 17 | 16 | 3ad2ant3 1136 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → 𝐿 = 𝐾) |
| 18 | bnj553.5 | . . . . 5 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)) | |
| 19 | bnj553.9 | . . . . 5 ⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) | |
| 20 | bnj553.4 | . . . . 5 ⊢ 𝐺 = (𝑓 ∪ {〈𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)〉}) | |
| 21 | 18, 19, 15, 20, 1 | bnj548 34911 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → 𝐵 = 𝐾) |
| 22 | 21 | 3adant3 1133 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → 𝐵 = 𝐾) |
| 23 | fveq2 6906 | . . . . . 6 ⊢ (𝑝 = 𝑖 → (𝑓‘𝑝) = (𝑓‘𝑖)) | |
| 24 | 23 | bnj1113 34799 | . . . . 5 ⊢ (𝑝 = 𝑖 → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
| 25 | bnj553.7 | . . . . . . 7 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) | |
| 26 | 19, 25 | eqeq12i 2755 | . . . . . 6 ⊢ (𝐵 = 𝐶 ↔ ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)) |
| 27 | eqcom 2744 | . . . . . 6 ⊢ (∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ↔ ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) | |
| 28 | 26, 27 | bitri 275 | . . . . 5 ⊢ (𝐵 = 𝐶 ↔ ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
| 29 | 24, 28 | sylibr 234 | . . . 4 ⊢ (𝑝 = 𝑖 → 𝐵 = 𝐶) |
| 30 | 29 | 3ad2ant3 1136 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → 𝐵 = 𝐶) |
| 31 | 17, 22, 30 | 3eqtr2rd 2784 | . 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → 𝐶 = 𝐿) |
| 32 | 11, 31 | eqtrd 2777 | 1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → (𝐺‘𝑚) = 𝐿) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∖ cdif 3948 ∪ cun 3949 ∅c0 4333 {csn 4626 〈cop 4632 ∪ ciun 4991 suc csuc 6386 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 ωcom 7887 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 |
| 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-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-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: bnj557 34915 |
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