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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1448 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 33674. 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 |
---|---|
bnj1448.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1448.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1448.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1448.4 | ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
bnj1448.5 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} |
bnj1448.6 | ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) |
bnj1448.7 | ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) |
bnj1448.8 | ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) |
bnj1448.9 | ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} |
bnj1448.10 | ⊢ 𝑃 = ∪ 𝐻 |
bnj1448.11 | ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1448.12 | ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) |
bnj1448.13 | ⊢ 𝑊 = 〈𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))〉 |
Ref | Expression |
---|---|
bnj1448 | ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑓(𝑄‘𝑧) = (𝐺‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1448.12 | . . . . 5 ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) | |
2 | bnj1448.10 | . . . . . . 7 ⊢ 𝑃 = ∪ 𝐻 | |
3 | bnj1448.9 | . . . . . . . . . 10 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} | |
4 | 3 | bnj1317 33433 | . . . . . . . . 9 ⊢ (𝑤 ∈ 𝐻 → ∀𝑓 𝑤 ∈ 𝐻) |
5 | 4 | nfcii 2891 | . . . . . . . 8 ⊢ Ⅎ𝑓𝐻 |
6 | 5 | nfuni 4872 | . . . . . . 7 ⊢ Ⅎ𝑓∪ 𝐻 |
7 | 2, 6 | nfcxfr 2905 | . . . . . 6 ⊢ Ⅎ𝑓𝑃 |
8 | nfcv 2907 | . . . . . . . 8 ⊢ Ⅎ𝑓𝑥 | |
9 | nfcv 2907 | . . . . . . . . 9 ⊢ Ⅎ𝑓𝐺 | |
10 | bnj1448.11 | . . . . . . . . . 10 ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
11 | nfcv 2907 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑓 pred(𝑥, 𝐴, 𝑅) | |
12 | 7, 11 | nfres 5939 | . . . . . . . . . . 11 ⊢ Ⅎ𝑓(𝑃 ↾ pred(𝑥, 𝐴, 𝑅)) |
13 | 8, 12 | nfop 4846 | . . . . . . . . . 10 ⊢ Ⅎ𝑓〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
14 | 10, 13 | nfcxfr 2905 | . . . . . . . . 9 ⊢ Ⅎ𝑓𝑍 |
15 | 9, 14 | nffv 6852 | . . . . . . . 8 ⊢ Ⅎ𝑓(𝐺‘𝑍) |
16 | 8, 15 | nfop 4846 | . . . . . . 7 ⊢ Ⅎ𝑓〈𝑥, (𝐺‘𝑍)〉 |
17 | 16 | nfsn 4668 | . . . . . 6 ⊢ Ⅎ𝑓{〈𝑥, (𝐺‘𝑍)〉} |
18 | 7, 17 | nfun 4125 | . . . . 5 ⊢ Ⅎ𝑓(𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) |
19 | 1, 18 | nfcxfr 2905 | . . . 4 ⊢ Ⅎ𝑓𝑄 |
20 | nfcv 2907 | . . . 4 ⊢ Ⅎ𝑓𝑧 | |
21 | 19, 20 | nffv 6852 | . . 3 ⊢ Ⅎ𝑓(𝑄‘𝑧) |
22 | bnj1448.13 | . . . . 5 ⊢ 𝑊 = 〈𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))〉 | |
23 | nfcv 2907 | . . . . . . 7 ⊢ Ⅎ𝑓 pred(𝑧, 𝐴, 𝑅) | |
24 | 19, 23 | nfres 5939 | . . . . . 6 ⊢ Ⅎ𝑓(𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) |
25 | 20, 24 | nfop 4846 | . . . . 5 ⊢ Ⅎ𝑓〈𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))〉 |
26 | 22, 25 | nfcxfr 2905 | . . . 4 ⊢ Ⅎ𝑓𝑊 |
27 | 9, 26 | nffv 6852 | . . 3 ⊢ Ⅎ𝑓(𝐺‘𝑊) |
28 | 21, 27 | nfeq 2920 | . 2 ⊢ Ⅎ𝑓(𝑄‘𝑧) = (𝐺‘𝑊) |
29 | 28 | nf5ri 2188 | 1 ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑓(𝑄‘𝑧) = (𝐺‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 ∀wal 1539 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2713 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 {crab 3407 [wsbc 3739 ∪ cun 3908 ⊆ wss 3910 ∅c0 4282 {csn 4586 〈cop 4592 ∪ cuni 4865 class class class wbr 5105 dom cdm 5633 ↾ cres 5635 Fn wfn 6491 ‘cfv 6496 predc-bnj14 33300 FrSe w-bnj15 33304 trClc-bnj18 33306 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-xp 5639 df-res 5645 df-iota 6448 df-fv 6504 |
This theorem is referenced by: bnj1450 33662 bnj1463 33667 |
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