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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1448 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 32336. 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 32095 | . . . . . . . . 9 ⊢ (𝑤 ∈ 𝐻 → ∀𝑓 𝑤 ∈ 𝐻) |
5 | 4 | nfcii 2967 | . . . . . . . 8 ⊢ Ⅎ𝑓𝐻 |
6 | 5 | nfuni 4847 | . . . . . . 7 ⊢ Ⅎ𝑓∪ 𝐻 |
7 | 2, 6 | nfcxfr 2977 | . . . . . 6 ⊢ Ⅎ𝑓𝑃 |
8 | nfcv 2979 | . . . . . . . 8 ⊢ Ⅎ𝑓𝑥 | |
9 | nfcv 2979 | . . . . . . . . 9 ⊢ Ⅎ𝑓𝐺 | |
10 | bnj1448.11 | . . . . . . . . . 10 ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
11 | nfcv 2979 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑓 pred(𝑥, 𝐴, 𝑅) | |
12 | 7, 11 | nfres 5857 | . . . . . . . . . . 11 ⊢ Ⅎ𝑓(𝑃 ↾ pred(𝑥, 𝐴, 𝑅)) |
13 | 8, 12 | nfop 4821 | . . . . . . . . . 10 ⊢ Ⅎ𝑓〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
14 | 10, 13 | nfcxfr 2977 | . . . . . . . . 9 ⊢ Ⅎ𝑓𝑍 |
15 | 9, 14 | nffv 6682 | . . . . . . . 8 ⊢ Ⅎ𝑓(𝐺‘𝑍) |
16 | 8, 15 | nfop 4821 | . . . . . . 7 ⊢ Ⅎ𝑓〈𝑥, (𝐺‘𝑍)〉 |
17 | 16 | nfsn 4645 | . . . . . 6 ⊢ Ⅎ𝑓{〈𝑥, (𝐺‘𝑍)〉} |
18 | 7, 17 | nfun 4143 | . . . . 5 ⊢ Ⅎ𝑓(𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) |
19 | 1, 18 | nfcxfr 2977 | . . . 4 ⊢ Ⅎ𝑓𝑄 |
20 | nfcv 2979 | . . . 4 ⊢ Ⅎ𝑓𝑧 | |
21 | 19, 20 | nffv 6682 | . . 3 ⊢ Ⅎ𝑓(𝑄‘𝑧) |
22 | bnj1448.13 | . . . . 5 ⊢ 𝑊 = 〈𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))〉 | |
23 | nfcv 2979 | . . . . . . 7 ⊢ Ⅎ𝑓 pred(𝑧, 𝐴, 𝑅) | |
24 | 19, 23 | nfres 5857 | . . . . . 6 ⊢ Ⅎ𝑓(𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) |
25 | 20, 24 | nfop 4821 | . . . . 5 ⊢ Ⅎ𝑓〈𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))〉 |
26 | 22, 25 | nfcxfr 2977 | . . . 4 ⊢ Ⅎ𝑓𝑊 |
27 | 9, 26 | nffv 6682 | . . 3 ⊢ Ⅎ𝑓(𝐺‘𝑊) |
28 | 21, 27 | nfeq 2993 | . 2 ⊢ Ⅎ𝑓(𝑄‘𝑧) = (𝐺‘𝑊) |
29 | 28 | nf5ri 2195 | 1 ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑓(𝑄‘𝑧) = (𝐺‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∀wal 1535 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2801 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 {crab 3144 [wsbc 3774 ∪ cun 3936 ⊆ wss 3938 ∅c0 4293 {csn 4569 〈cop 4575 ∪ cuni 4840 class class class wbr 5068 dom cdm 5557 ↾ cres 5559 Fn wfn 6352 ‘cfv 6357 predc-bnj14 31960 FrSe w-bnj15 31964 trClc-bnj18 31966 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-res 5569 df-iota 6316 df-fv 6365 |
This theorem is referenced by: bnj1450 32324 bnj1463 32329 |
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