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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1466 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 32444. 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 |
---|---|
bnj1466.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1466.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1466.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1466.4 | ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
bnj1466.5 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} |
bnj1466.6 | ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) |
bnj1466.7 | ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) |
bnj1466.8 | ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) |
bnj1466.9 | ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} |
bnj1466.10 | ⊢ 𝑃 = ∪ 𝐻 |
bnj1466.11 | ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1466.12 | ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) |
Ref | Expression |
---|---|
bnj1466 | ⊢ (𝑤 ∈ 𝑄 → ∀𝑓 𝑤 ∈ 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1466.12 | . . 3 ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) | |
2 | bnj1466.10 | . . . . 5 ⊢ 𝑃 = ∪ 𝐻 | |
3 | bnj1466.9 | . . . . . . . 8 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} | |
4 | 3 | bnj1317 32203 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐻 → ∀𝑓 𝑤 ∈ 𝐻) |
5 | 4 | nfcii 2940 | . . . . . 6 ⊢ Ⅎ𝑓𝐻 |
6 | 5 | nfuni 4807 | . . . . 5 ⊢ Ⅎ𝑓∪ 𝐻 |
7 | 2, 6 | nfcxfr 2953 | . . . 4 ⊢ Ⅎ𝑓𝑃 |
8 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑓𝑥 | |
9 | nfcv 2955 | . . . . . . 7 ⊢ Ⅎ𝑓𝐺 | |
10 | bnj1466.11 | . . . . . . . 8 ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
11 | nfcv 2955 | . . . . . . . . . 10 ⊢ Ⅎ𝑓 pred(𝑥, 𝐴, 𝑅) | |
12 | 7, 11 | nfres 5820 | . . . . . . . . 9 ⊢ Ⅎ𝑓(𝑃 ↾ pred(𝑥, 𝐴, 𝑅)) |
13 | 8, 12 | nfop 4781 | . . . . . . . 8 ⊢ Ⅎ𝑓〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
14 | 10, 13 | nfcxfr 2953 | . . . . . . 7 ⊢ Ⅎ𝑓𝑍 |
15 | 9, 14 | nffv 6655 | . . . . . 6 ⊢ Ⅎ𝑓(𝐺‘𝑍) |
16 | 8, 15 | nfop 4781 | . . . . 5 ⊢ Ⅎ𝑓〈𝑥, (𝐺‘𝑍)〉 |
17 | 16 | nfsn 4603 | . . . 4 ⊢ Ⅎ𝑓{〈𝑥, (𝐺‘𝑍)〉} |
18 | 7, 17 | nfun 4092 | . . 3 ⊢ Ⅎ𝑓(𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) |
19 | 1, 18 | nfcxfr 2953 | . 2 ⊢ Ⅎ𝑓𝑄 |
20 | 19 | nfcrii 2948 | 1 ⊢ (𝑤 ∈ 𝑄 → ∀𝑓 𝑤 ∈ 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 ∀wal 1536 = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2776 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 {crab 3110 [wsbc 3720 ∪ cun 3879 ⊆ wss 3881 ∅c0 4243 {csn 4525 〈cop 4531 ∪ cuni 4800 class class class wbr 5030 dom cdm 5519 ↾ cres 5521 Fn wfn 6319 ‘cfv 6324 predc-bnj14 32068 FrSe w-bnj15 32072 trClc-bnj18 32074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-res 5531 df-iota 6283 df-fv 6332 |
This theorem is referenced by: bnj1463 32437 bnj1491 32439 |
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