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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj910 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32286. 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 32223 | . . 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 32222 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V) |
10 | simpr3 1192 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑝 = suc 𝑛) | |
11 | 1 | bnj1235 32080 | . . . . . 6 ⊢ (𝜒 → 𝑓 Fn 𝑛) |
12 | 11 | 3ad2ant1 1129 | . . . . 5 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑓 Fn 𝑛) |
13 | 12 | adantl 484 | . . . 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 32048 | . . . . 5 ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)) |
16 | 15 | 3impib 1112 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝) |
17 | 9, 10, 13, 16 | syl3anc 1367 | . . 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 32214 | . . 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 32221 | . . . 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 32220 | . . . 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 32219 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜓″) |
26 | 3, 17, 20, 25 | bnj951 32051 | . 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 3500 | . . . 4 ⊢ 𝑝 ∈ V | |
29 | 1, 18, 21, 27, 28 | bnj919 32042 | . . 3 ⊢ (𝜒′ ↔ (𝑝 ∈ 𝐷 ∧ 𝑓 Fn 𝑝 ∧ 𝜑′ ∧ 𝜓′)) |
30 | bnj910.9 | . . 3 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′) | |
31 | 14 | bnj918 32041 | . . 3 ⊢ 𝐺 ∈ V |
32 | 29, 19, 22, 30, 31 | bnj976 32053 | . 2 ⊢ (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″)) |
33 | 26, 32 | sylibr 236 | 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 {cab 2802 ∀wral 3141 ∃wrex 3142 Vcvv 3497 [wsbc 3775 ∖ cdif 3936 ∪ cun 3937 ∅c0 4294 {csn 4570 〈cop 4576 ∪ ciun 4922 suc csuc 6196 Fn wfn 6353 ‘cfv 6358 ωcom 7583 ∧ w-bnj17 31960 predc-bnj14 31962 FrSe w-bnj15 31966 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-un 7464 ax-reg 9059 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7584 df-bnj17 31961 df-bnj14 31963 df-bnj13 31965 df-bnj15 31967 |
This theorem is referenced by: bnj998 32233 |
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