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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj546 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj852 34913. 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 |
|---|---|
| bnj546.1 | ⊢ 𝐷 = (ω ∖ {∅}) |
| bnj546.2 | ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)) |
| bnj546.3 | ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚)) |
| bnj546.4 | ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
| bnj546.5 | ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
| Ref | Expression |
|---|---|
| bnj546 | ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj546.2 | . . . . . . 7 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)) | |
| 2 | 3simpc 1151 | . . . . . . 7 ⊢ ((𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′) → (𝜑′ ∧ 𝜓′)) | |
| 3 | 1, 2 | sylbi 217 | . . . . . 6 ⊢ (𝜏 → (𝜑′ ∧ 𝜓′)) |
| 4 | bnj546.3 | . . . . . . 7 ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚)) | |
| 5 | bnj546.1 | . . . . . . . . . 10 ⊢ 𝐷 = (ω ∖ {∅}) | |
| 6 | 5 | bnj923 34760 | . . . . . . . . 9 ⊢ (𝑚 ∈ 𝐷 → 𝑚 ∈ ω) |
| 7 | 6 | 3ad2ant1 1134 | . . . . . . . 8 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) → 𝑚 ∈ ω) |
| 8 | simp3 1139 | . . . . . . . 8 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) → 𝑝 ∈ 𝑚) | |
| 9 | 7, 8 | jca 511 | . . . . . . 7 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) → (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)) |
| 10 | 4, 9 | sylbi 217 | . . . . . 6 ⊢ (𝜎 → (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)) |
| 11 | 3, 10 | anim12i 613 | . . . . 5 ⊢ ((𝜏 ∧ 𝜎) → ((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))) |
| 12 | bnj256 34698 | . . . . 5 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ ((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))) | |
| 13 | 11, 12 | sylibr 234 | . . . 4 ⊢ ((𝜏 ∧ 𝜎) → (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)) |
| 14 | 13 | anim2i 617 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜏 ∧ 𝜎)) → (𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))) |
| 15 | 14 | 3impb 1115 | . 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))) |
| 16 | bnj546.4 | . . 3 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | |
| 17 | bnj546.5 | . . 3 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
| 18 | biid 261 | . . 3 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)) | |
| 19 | 16, 17, 18 | bnj518 34878 | . 2 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) |
| 20 | fvex 6917 | . . 3 ⊢ (𝑓‘𝑝) ∈ V | |
| 21 | iunexg 7984 | . . 3 ⊢ (((𝑓‘𝑝) ∈ V ∧ ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) | |
| 22 | 20, 21 | mpan 690 | . 2 ⊢ (∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) |
| 23 | 15, 19, 22 | 3syl 18 | 1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3060 Vcvv 3479 ∖ cdif 3947 ∅c0 4332 {csn 4624 ∪ ciun 4989 suc csuc 6384 Fn wfn 6554 ‘cfv 6559 ωcom 7883 ∧ w-bnj17 34678 predc-bnj14 34680 FrSe w-bnj15 34684 |
| 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-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pr 5430 ax-un 7751 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-tr 5258 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fv 6567 df-om 7884 df-bnj17 34679 df-bnj14 34681 df-bnj13 34683 df-bnj15 34685 |
| This theorem is referenced by: bnj938 34929 |
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