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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj910 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 33679. 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 |
---|---|
bnj910.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
bnj910.2 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
bnj910.3 | ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
bnj910.4 | ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑) |
bnj910.5 | ⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓) |
bnj910.6 | ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒) |
bnj910.7 | ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′) |
bnj910.8 | ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′) |
bnj910.9 | ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′) |
bnj910.10 | ⊢ 𝐷 = (ω ∖ {∅}) |
bnj910.11 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
bnj910.12 | ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) |
bnj910.13 | ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩}) |
bnj910.14 | ⊢ (𝜏 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
bnj910.15 | ⊢ (𝜎 ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛)) |
Ref | Expression |
---|---|
bnj910 | ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜒″) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj910.3 | . . . 4 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
2 | bnj910.10 | . . . 4 ⊢ 𝐷 = (ω ∖ {∅}) | |
3 | 1, 2 | bnj970 33616 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑝 ∈ 𝐷) |
4 | bnj910.1 | . . . . 5 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
5 | bnj910.2 | . . . . 5 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
6 | bnj910.12 | . . . . 5 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) | |
7 | bnj910.14 | . . . . 5 ⊢ (𝜏 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
8 | bnj910.15 | . . . . 5 ⊢ (𝜎 ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛)) | |
9 | 4, 5, 1, 2, 6, 7, 8 | bnj969 33615 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V) |
10 | simpr3 1197 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑝 = suc 𝑛) | |
11 | 1 | bnj1235 33473 | . . . . . 6 ⊢ (𝜒 → 𝑓 Fn 𝑛) |
12 | 11 | 3ad2ant1 1134 | . . . . 5 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑓 Fn 𝑛) |
13 | 12 | adantl 483 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑓 Fn 𝑛) |
14 | bnj910.13 | . . . . . 6 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩}) | |
15 | 14 | bnj941 33441 | . . . . 5 ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)) |
16 | 15 | 3impib 1117 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝) |
17 | 9, 10, 13, 16 | syl3anc 1372 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐺 Fn 𝑝) |
18 | bnj910.4 | . . . 4 ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑) | |
19 | bnj910.7 | . . . 4 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′) | |
20 | 4, 5, 1, 18, 19, 2, 6, 14, 7, 8 | bnj944 33607 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜑″) |
21 | bnj910.5 | . . . 4 ⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓) | |
22 | bnj910.8 | . . . 4 ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′) | |
23 | 5, 1, 2, 6, 14, 9 | bnj967 33614 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
24 | 1, 2, 6, 14, 9, 17 | bnj966 33613 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
25 | 5, 1, 21, 22, 6, 14, 23, 24 | bnj964 33612 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜓″) |
26 | 3, 17, 20, 25 | bnj951 33444 | . 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″)) |
27 | bnj910.6 | . . . 4 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒) | |
28 | vex 3448 | . . . 4 ⊢ 𝑝 ∈ V | |
29 | 1, 18, 21, 27, 28 | bnj919 33436 | . . 3 ⊢ (𝜒′ ↔ (𝑝 ∈ 𝐷 ∧ 𝑓 Fn 𝑝 ∧ 𝜑′ ∧ 𝜓′)) |
30 | bnj910.9 | . . 3 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′) | |
31 | 14 | bnj918 33435 | . . 3 ⊢ 𝐺 ∈ V |
32 | 29, 19, 22, 30, 31 | bnj976 33446 | . 2 ⊢ (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″)) |
33 | 26, 32 | sylibr 233 | 1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜒″) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3061 ∃wrex 3070 Vcvv 3444 [wsbc 3740 ∖ cdif 3908 ∪ cun 3909 ∅c0 4283 {csn 4587 ⟨cop 4593 ∪ ciun 4955 suc csuc 6320 Fn wfn 6492 ‘cfv 6497 ωcom 7803 ∧ w-bnj17 33355 predc-bnj14 33357 FrSe w-bnj15 33361 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 ax-reg 9533 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-res 5646 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 df-om 7804 df-bnj17 33356 df-bnj14 33358 df-bnj13 33360 df-bnj15 33362 |
This theorem is referenced by: bnj998 33626 |
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