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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1467 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj60 35077. 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 | 
|---|---|
| bnj1467.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} | 
| bnj1467.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 | 
| bnj1467.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | 
| bnj1467.4 | ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) | 
| bnj1467.5 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} | 
| bnj1467.6 | ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) | 
| bnj1467.7 | ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) | 
| bnj1467.8 | ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) | 
| bnj1467.9 | ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} | 
| bnj1467.10 | ⊢ 𝑃 = ∪ 𝐻 | 
| bnj1467.11 | ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 | 
| bnj1467.12 | ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) | 
| Ref | Expression | 
|---|---|
| bnj1467 | ⊢ (𝑤 ∈ 𝑄 → ∀𝑑 𝑤 ∈ 𝑄) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bnj1467.12 | . . 3 ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) | |
| 2 | bnj1467.10 | . . . . 5 ⊢ 𝑃 = ∪ 𝐻 | |
| 3 | bnj1467.9 | . . . . . . 7 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} | |
| 4 | nfcv 2904 | . . . . . . . . 9 ⊢ Ⅎ𝑑 pred(𝑥, 𝐴, 𝑅) | |
| 5 | bnj1467.8 | . . . . . . . . . 10 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) | |
| 6 | nfcv 2904 | . . . . . . . . . . 11 ⊢ Ⅎ𝑑𝑦 | |
| 7 | bnj1467.4 | . . . . . . . . . . . 12 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) | |
| 8 | bnj1467.3 | . . . . . . . . . . . . . . 15 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
| 9 | nfre1 3284 | . . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑑∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) | |
| 10 | 9 | nfab 2910 | . . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | 
| 11 | 8, 10 | nfcxfr 2902 | . . . . . . . . . . . . . 14 ⊢ Ⅎ𝑑𝐶 | 
| 12 | 11 | nfcri 2896 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑑 𝑓 ∈ 𝐶 | 
| 13 | nfv 1913 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑑dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) | |
| 14 | 12, 13 | nfan 1898 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑑(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) | 
| 15 | 7, 14 | nfxfr 1852 | . . . . . . . . . . 11 ⊢ Ⅎ𝑑𝜏 | 
| 16 | 6, 15 | nfsbcw 3809 | . . . . . . . . . 10 ⊢ Ⅎ𝑑[𝑦 / 𝑥]𝜏 | 
| 17 | 5, 16 | nfxfr 1852 | . . . . . . . . 9 ⊢ Ⅎ𝑑𝜏′ | 
| 18 | 4, 17 | nfrexw 3312 | . . . . . . . 8 ⊢ Ⅎ𝑑∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ | 
| 19 | 18 | nfab 2910 | . . . . . . 7 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} | 
| 20 | 3, 19 | nfcxfr 2902 | . . . . . 6 ⊢ Ⅎ𝑑𝐻 | 
| 21 | 20 | nfuni 4913 | . . . . 5 ⊢ Ⅎ𝑑∪ 𝐻 | 
| 22 | 2, 21 | nfcxfr 2902 | . . . 4 ⊢ Ⅎ𝑑𝑃 | 
| 23 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑑𝑥 | |
| 24 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑑𝐺 | |
| 25 | bnj1467.11 | . . . . . . . 8 ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
| 26 | 22, 4 | nfres 5998 | . . . . . . . . 9 ⊢ Ⅎ𝑑(𝑃 ↾ pred(𝑥, 𝐴, 𝑅)) | 
| 27 | 23, 26 | nfop 4888 | . . . . . . . 8 ⊢ Ⅎ𝑑〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 | 
| 28 | 25, 27 | nfcxfr 2902 | . . . . . . 7 ⊢ Ⅎ𝑑𝑍 | 
| 29 | 24, 28 | nffv 6915 | . . . . . 6 ⊢ Ⅎ𝑑(𝐺‘𝑍) | 
| 30 | 23, 29 | nfop 4888 | . . . . 5 ⊢ Ⅎ𝑑〈𝑥, (𝐺‘𝑍)〉 | 
| 31 | 30 | nfsn 4706 | . . . 4 ⊢ Ⅎ𝑑{〈𝑥, (𝐺‘𝑍)〉} | 
| 32 | 22, 31 | nfun 4169 | . . 3 ⊢ Ⅎ𝑑(𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) | 
| 33 | 1, 32 | nfcxfr 2902 | . 2 ⊢ Ⅎ𝑑𝑄 | 
| 34 | 33 | nfcrii 2899 | 1 ⊢ (𝑤 ∈ 𝑄 → ∀𝑑 𝑤 ∈ 𝑄) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∀wal 1537 = wceq 1539 ∃wex 1778 ∈ wcel 2107 {cab 2713 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3435 [wsbc 3787 ∪ cun 3948 ⊆ wss 3950 ∅c0 4332 {csn 4625 〈cop 4631 ∪ cuni 4906 class class class wbr 5142 dom cdm 5684 ↾ cres 5686 Fn wfn 6555 ‘cfv 6560 predc-bnj14 34703 FrSe w-bnj15 34707 trClc-bnj18 34709 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-res 5696 df-iota 6513 df-fv 6568 | 
| This theorem is referenced by: bnj1463 35070 | 
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