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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1415 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 30384. 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 |
---|---|
bnj1415.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1415.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1415.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1415.4 | ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
bnj1415.5 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} |
bnj1415.6 | ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) |
bnj1415.7 | ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) |
bnj1415.8 | ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) |
bnj1415.9 | ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} |
bnj1415.10 | ⊢ 𝑃 = ∪ 𝐻 |
Ref | Expression |
---|---|
bnj1415 | ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1415.7 | . . . 4 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) | |
2 | bnj1415.6 | . . . . 5 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) | |
3 | 2 | simplbi 475 | . . . 4 ⊢ (𝜓 → 𝑅 FrSe 𝐴) |
4 | 1, 3 | bnj835 30083 | . . 3 ⊢ (𝜒 → 𝑅 FrSe 𝐴) |
5 | bnj1415.5 | . . . 4 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} | |
6 | 5, 1 | bnj1212 30124 | . . 3 ⊢ (𝜒 → 𝑥 ∈ 𝐴) |
7 | eqid 2610 | . . . 4 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) = ( pred(𝑥, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) | |
8 | 7 | bnj1414 30359 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → trCl(𝑥, 𝐴, 𝑅) = ( pred(𝑥, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))) |
9 | 4, 6, 8 | syl2anc 691 | . 2 ⊢ (𝜒 → trCl(𝑥, 𝐴, 𝑅) = ( pred(𝑥, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))) |
10 | iunun 4540 | . . . 4 ⊢ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = (∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅){𝑦} ∪ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) | |
11 | iunid 4511 | . . . . 5 ⊢ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅){𝑦} = pred(𝑥, 𝐴, 𝑅) | |
12 | 11 | uneq1i 3725 | . . . 4 ⊢ (∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅){𝑦} ∪ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) = ( pred(𝑥, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) |
13 | 10, 12 | eqtri 2632 | . . 3 ⊢ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = ( pred(𝑥, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) |
14 | bnj1415.1 | . . . 4 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} | |
15 | bnj1415.2 | . . . 4 ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
16 | bnj1415.3 | . . . 4 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
17 | bnj1415.4 | . . . 4 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) | |
18 | bnj1415.8 | . . . 4 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) | |
19 | bnj1415.9 | . . . 4 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} | |
20 | bnj1415.10 | . . . 4 ⊢ 𝑃 = ∪ 𝐻 | |
21 | biid 250 | . . . 4 ⊢ ((𝜒 ∧ 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ↔ (𝜒 ∧ 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))) | |
22 | biid 250 | . . . 4 ⊢ (((𝜒 ∧ 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ↔ ((𝜒 ∧ 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))) | |
23 | 14, 15, 16, 17, 5, 2, 1, 18, 19, 20, 21, 22 | bnj1398 30356 | . . 3 ⊢ (𝜒 → ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = dom 𝑃) |
24 | 13, 23 | syl5eqr 2658 | . 2 ⊢ (𝜒 → ( pred(𝑥, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) = dom 𝑃) |
25 | 9, 24 | eqtr2d 2645 | 1 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 [wsbc 3402 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 {csn 4125 〈cop 4131 ∪ cuni 4372 ∪ ciun 4455 class class class wbr 4583 dom cdm 5038 ↾ cres 5040 Fn wfn 5799 ‘cfv 5804 predc-bnj14 30007 FrSe w-bnj15 30011 trClc-bnj18 30013 |
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-pow 4769 ax-pr 4833 ax-un 6847 ax-reg 8380 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 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-1o 7447 df-bnj17 30006 df-bnj14 30008 df-bnj13 30010 df-bnj15 30012 df-bnj18 30014 df-bnj19 30016 |
This theorem is referenced by: bnj1312 30380 |
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