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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1467 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 32709. 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 2897 | . . . . . . . . 9 ⊢ Ⅎ𝑑 pred(𝑥, 𝐴, 𝑅) | |
5 | bnj1467.8 | . . . . . . . . . 10 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) | |
6 | nfcv 2897 | . . . . . . . . . . 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 3215 | . . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑑∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) | |
10 | 9 | nfab 2903 | . . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
11 | 8, 10 | nfcxfr 2895 | . . . . . . . . . . . . . 14 ⊢ Ⅎ𝑑𝐶 |
12 | 11 | nfcri 2884 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑑 𝑓 ∈ 𝐶 |
13 | nfv 1922 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑑dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) | |
14 | 12, 13 | nfan 1907 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑑(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) |
15 | 7, 14 | nfxfr 1860 | . . . . . . . . . . 11 ⊢ Ⅎ𝑑𝜏 |
16 | 6, 15 | nfsbcw 3705 | . . . . . . . . . 10 ⊢ Ⅎ𝑑[𝑦 / 𝑥]𝜏 |
17 | 5, 16 | nfxfr 1860 | . . . . . . . . 9 ⊢ Ⅎ𝑑𝜏′ |
18 | 4, 17 | nfrex 3218 | . . . . . . . 8 ⊢ Ⅎ𝑑∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ |
19 | 18 | nfab 2903 | . . . . . . 7 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} |
20 | 3, 19 | nfcxfr 2895 | . . . . . 6 ⊢ Ⅎ𝑑𝐻 |
21 | 20 | nfuni 4812 | . . . . 5 ⊢ Ⅎ𝑑∪ 𝐻 |
22 | 2, 21 | nfcxfr 2895 | . . . 4 ⊢ Ⅎ𝑑𝑃 |
23 | nfcv 2897 | . . . . . 6 ⊢ Ⅎ𝑑𝑥 | |
24 | nfcv 2897 | . . . . . . 7 ⊢ Ⅎ𝑑𝐺 | |
25 | bnj1467.11 | . . . . . . . 8 ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
26 | 22, 4 | nfres 5838 | . . . . . . . . 9 ⊢ Ⅎ𝑑(𝑃 ↾ pred(𝑥, 𝐴, 𝑅)) |
27 | 23, 26 | nfop 4786 | . . . . . . . 8 ⊢ Ⅎ𝑑〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
28 | 25, 27 | nfcxfr 2895 | . . . . . . 7 ⊢ Ⅎ𝑑𝑍 |
29 | 24, 28 | nffv 6705 | . . . . . 6 ⊢ Ⅎ𝑑(𝐺‘𝑍) |
30 | 23, 29 | nfop 4786 | . . . . 5 ⊢ Ⅎ𝑑〈𝑥, (𝐺‘𝑍)〉 |
31 | 30 | nfsn 4609 | . . . 4 ⊢ Ⅎ𝑑{〈𝑥, (𝐺‘𝑍)〉} |
32 | 22, 31 | nfun 4065 | . . 3 ⊢ Ⅎ𝑑(𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) |
33 | 1, 32 | nfcxfr 2895 | . 2 ⊢ Ⅎ𝑑𝑄 |
34 | 33 | nfcrii 2889 | 1 ⊢ (𝑤 ∈ 𝑄 → ∀𝑑 𝑤 ∈ 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 ∀wal 1541 = wceq 1543 ∃wex 1787 ∈ wcel 2112 {cab 2714 ≠ wne 2932 ∀wral 3051 ∃wrex 3052 {crab 3055 [wsbc 3683 ∪ cun 3851 ⊆ wss 3853 ∅c0 4223 {csn 4527 〈cop 4533 ∪ cuni 4805 class class class wbr 5039 dom cdm 5536 ↾ cres 5538 Fn wfn 6353 ‘cfv 6358 predc-bnj14 32333 FrSe w-bnj15 32337 trClc-bnj18 32339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-xp 5542 df-res 5548 df-iota 6316 df-fv 6366 |
This theorem is referenced by: bnj1463 32702 |
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