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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj910 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 30332. 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 30271 | . . 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 30270 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V) |
10 | simpr3 1062 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑝 = suc 𝑛) | |
11 | 1 | bnj1235 30129 | . . . . . 6 ⊢ (𝜒 → 𝑓 Fn 𝑛) |
12 | 11 | 3ad2ant1 1075 | . . . . 5 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑓 Fn 𝑛) |
13 | 12 | adantl 481 | . . . 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 30097 | . . . . 5 ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)) |
16 | 15 | 3impib 1254 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝) |
17 | 9, 10, 13, 16 | syl3anc 1318 | . . 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 30262 | . . 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 30269 | . . . 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 30268 | . . . 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 30267 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜓″) |
26 | 3, 17, 20, 25 | bnj951 30100 | . 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 3176 | . . . 4 ⊢ 𝑝 ∈ V | |
29 | 1, 18, 21, 27, 28 | bnj919 30091 | . . 3 ⊢ (𝜒′ ↔ (𝑝 ∈ 𝐷 ∧ 𝑓 Fn 𝑝 ∧ 𝜑′ ∧ 𝜓′)) |
30 | bnj910.9 | . . 3 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′) | |
31 | 14 | bnj918 30090 | . . 3 ⊢ 𝐺 ∈ V |
32 | 29, 19, 22, 30, 31 | bnj976 30102 | . 2 ⊢ (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″)) |
33 | 26, 32 | sylibr 223 | 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {cab 2596 ∀wral 2896 ∃wrex 2897 Vcvv 3173 [wsbc 3402 ∖ cdif 3537 ∪ cun 3538 ∅c0 3874 {csn 4125 〈cop 4131 ∪ ciun 4455 suc csuc 5642 Fn wfn 5799 ‘cfv 5804 ωcom 6957 ∧ w-bnj17 30005 predc-bnj14 30007 FrSe w-bnj15 30011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 ax-reg 8380 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-bnj17 30006 df-bnj14 30008 df-bnj13 30010 df-bnj15 30012 |
This theorem is referenced by: bnj998 30280 |
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