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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj864 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 33662. 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 |
---|---|
bnj864.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
bnj864.2 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
bnj864.3 | ⊢ 𝐷 = (ω ∖ {∅}) |
bnj864.4 | ⊢ (𝜒 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷)) |
bnj864.5 | ⊢ (𝜃 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
Ref | Expression |
---|---|
bnj864 | ⊢ (𝜒 → ∃!𝑓𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj864.1 | . . . . 5 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
2 | bnj864.2 | . . . . 5 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
3 | bnj864.3 | . . . . 5 ⊢ 𝐷 = (ω ∖ {∅}) | |
4 | 1, 2, 3 | bnj852 33573 | . . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
5 | df-ral 3066 | . . . . . 6 ⊢ (∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∀𝑛(𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) | |
6 | 5 | imbi2i 336 | . . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛(𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))) |
7 | 19.21v 1943 | . . . . 5 ⊢ (∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛(𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))) | |
8 | impexp 452 | . . . . . . 7 ⊢ ((((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))) | |
9 | df-3an 1090 | . . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷)) | |
10 | 9 | bicomi 223 | . . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷) ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷)) |
11 | 10 | imbi1i 350 | . . . . . . 7 ⊢ ((((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) |
12 | 8, 11 | bitr3i 277 | . . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) |
13 | 12 | albii 1822 | . . . . 5 ⊢ (∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) ↔ ∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) |
14 | 6, 7, 13 | 3bitr2i 299 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) |
15 | 4, 14 | mpbi 229 | . . 3 ⊢ ∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
16 | 15 | spi 2178 | . 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
17 | bnj864.4 | . 2 ⊢ (𝜒 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷)) | |
18 | bnj864.5 | . . 3 ⊢ (𝜃 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
19 | 18 | eubii 2584 | . 2 ⊢ (∃!𝑓𝜃 ↔ ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
20 | 16, 17, 19 | 3imtr4i 292 | 1 ⊢ (𝜒 → ∃!𝑓𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∀wal 1540 = wceq 1542 ∈ wcel 2107 ∃!weu 2567 ∀wral 3065 ∖ cdif 3912 ∅c0 4287 {csn 4591 ∪ ciun 4959 suc csuc 6324 Fn wfn 6496 ‘cfv 6501 ωcom 7807 predc-bnj14 33340 FrSe w-bnj15 33344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-reg 9535 ax-inf2 9584 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-om 7808 df-1o 8417 df-bnj17 33339 df-bnj14 33341 df-bnj13 33343 df-bnj15 33345 |
This theorem is referenced by: bnj849 33577 |
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