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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1447 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 34372. 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 |
---|---|
bnj1447.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1447.2 | ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ |
bnj1447.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1447.4 | ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
bnj1447.5 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} |
bnj1447.6 | ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) |
bnj1447.7 | ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) |
bnj1447.8 | ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) |
bnj1447.9 | ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} |
bnj1447.10 | ⊢ 𝑃 = ∪ 𝐻 |
bnj1447.11 | ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩ |
bnj1447.12 | ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}) |
bnj1447.13 | ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ |
Ref | Expression |
---|---|
bnj1447 | ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑦(𝑄‘𝑧) = (𝐺‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1447.12 | . . . . 5 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}) | |
2 | bnj1447.10 | . . . . . . 7 ⊢ 𝑃 = ∪ 𝐻 | |
3 | bnj1447.9 | . . . . . . . . 9 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} | |
4 | nfre1 3281 | . . . . . . . . . 10 ⊢ Ⅎ𝑦∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ | |
5 | 4 | nfab 2908 | . . . . . . . . 9 ⊢ Ⅎ𝑦{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} |
6 | 3, 5 | nfcxfr 2900 | . . . . . . . 8 ⊢ Ⅎ𝑦𝐻 |
7 | 6 | nfuni 4915 | . . . . . . 7 ⊢ Ⅎ𝑦∪ 𝐻 |
8 | 2, 7 | nfcxfr 2900 | . . . . . 6 ⊢ Ⅎ𝑦𝑃 |
9 | nfcv 2902 | . . . . . . . 8 ⊢ Ⅎ𝑦𝑥 | |
10 | nfcv 2902 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝐺 | |
11 | bnj1447.11 | . . . . . . . . . 10 ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩ | |
12 | nfcv 2902 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑦 pred(𝑥, 𝐴, 𝑅) | |
13 | 8, 12 | nfres 5983 | . . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑃 ↾ pred(𝑥, 𝐴, 𝑅)) |
14 | 9, 13 | nfop 4889 | . . . . . . . . . 10 ⊢ Ⅎ𝑦⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩ |
15 | 11, 14 | nfcxfr 2900 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝑍 |
16 | 10, 15 | nffv 6901 | . . . . . . . 8 ⊢ Ⅎ𝑦(𝐺‘𝑍) |
17 | 9, 16 | nfop 4889 | . . . . . . 7 ⊢ Ⅎ𝑦⟨𝑥, (𝐺‘𝑍)⟩ |
18 | 17 | nfsn 4711 | . . . . . 6 ⊢ Ⅎ𝑦{⟨𝑥, (𝐺‘𝑍)⟩} |
19 | 8, 18 | nfun 4165 | . . . . 5 ⊢ Ⅎ𝑦(𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}) |
20 | 1, 19 | nfcxfr 2900 | . . . 4 ⊢ Ⅎ𝑦𝑄 |
21 | nfcv 2902 | . . . 4 ⊢ Ⅎ𝑦𝑧 | |
22 | 20, 21 | nffv 6901 | . . 3 ⊢ Ⅎ𝑦(𝑄‘𝑧) |
23 | bnj1447.13 | . . . . 5 ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ | |
24 | nfcv 2902 | . . . . . . 7 ⊢ Ⅎ𝑦 pred(𝑧, 𝐴, 𝑅) | |
25 | 20, 24 | nfres 5983 | . . . . . 6 ⊢ Ⅎ𝑦(𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) |
26 | 21, 25 | nfop 4889 | . . . . 5 ⊢ Ⅎ𝑦⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ |
27 | 23, 26 | nfcxfr 2900 | . . . 4 ⊢ Ⅎ𝑦𝑊 |
28 | 10, 27 | nffv 6901 | . . 3 ⊢ Ⅎ𝑦(𝐺‘𝑊) |
29 | 22, 28 | nfeq 2915 | . 2 ⊢ Ⅎ𝑦(𝑄‘𝑧) = (𝐺‘𝑊) |
30 | 29 | nf5ri 2187 | 1 ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑦(𝑄‘𝑧) = (𝐺‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 ∀wal 1538 = wceq 1540 ∃wex 1780 ∈ wcel 2105 {cab 2708 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3431 [wsbc 3777 ∪ cun 3946 ⊆ wss 3948 ∅c0 4322 {csn 4628 ⟨cop 4634 ∪ cuni 4908 class class class wbr 5148 dom cdm 5676 ↾ cres 5678 Fn wfn 6538 ‘cfv 6543 predc-bnj14 33998 FrSe w-bnj15 34002 trClc-bnj18 34004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5682 df-res 5688 df-iota 6495 df-fv 6551 |
This theorem is referenced by: bnj1450 34360 |
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