Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1491 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 30384. 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 |
---|---|
bnj1491.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1491.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1491.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1491.4 | ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
bnj1491.5 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} |
bnj1491.6 | ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) |
bnj1491.7 | ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) |
bnj1491.8 | ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) |
bnj1491.9 | ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} |
bnj1491.10 | ⊢ 𝑃 = ∪ 𝐻 |
bnj1491.11 | ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1491.12 | ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) |
bnj1491.13 | ⊢ (𝜒 → (𝑄 ∈ 𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
Ref | Expression |
---|---|
bnj1491 | ⊢ ((𝜒 ∧ 𝑄 ∈ V) → ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1491.13 | . 2 ⊢ (𝜒 → (𝑄 ∈ 𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) | |
2 | bnj1491.1 | . . . . 5 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} | |
3 | bnj1491.2 | . . . . 5 ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
4 | bnj1491.3 | . . . . 5 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
5 | bnj1491.4 | . . . . 5 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) | |
6 | bnj1491.5 | . . . . 5 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} | |
7 | bnj1491.6 | . . . . 5 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) | |
8 | bnj1491.7 | . . . . 5 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) | |
9 | bnj1491.8 | . . . . 5 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) | |
10 | bnj1491.9 | . . . . 5 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} | |
11 | bnj1491.10 | . . . . 5 ⊢ 𝑃 = ∪ 𝐻 | |
12 | bnj1491.11 | . . . . 5 ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
13 | bnj1491.12 | . . . . 5 ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) | |
14 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | bnj1466 30375 | . . . 4 ⊢ (𝑤 ∈ 𝑄 → ∀𝑓 𝑤 ∈ 𝑄) |
15 | 14 | nfcii 2742 | . . 3 ⊢ Ⅎ𝑓𝑄 |
16 | 4 | bnj1317 30146 | . . . . . 6 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶) |
17 | 16 | nfcii 2742 | . . . . 5 ⊢ Ⅎ𝑓𝐶 |
18 | 15, 17 | nfel 2763 | . . . 4 ⊢ Ⅎ𝑓 𝑄 ∈ 𝐶 |
19 | 15 | nfdm 5288 | . . . . 5 ⊢ Ⅎ𝑓dom 𝑄 |
20 | 19 | nfeq1 2764 | . . . 4 ⊢ Ⅎ𝑓dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) |
21 | 18, 20 | nfan 1816 | . . 3 ⊢ Ⅎ𝑓(𝑄 ∈ 𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) |
22 | eleq1 2676 | . . . 4 ⊢ (𝑓 = 𝑄 → (𝑓 ∈ 𝐶 ↔ 𝑄 ∈ 𝐶)) | |
23 | dmeq 5246 | . . . . 5 ⊢ (𝑓 = 𝑄 → dom 𝑓 = dom 𝑄) | |
24 | 23 | eqeq1d 2612 | . . . 4 ⊢ (𝑓 = 𝑄 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
25 | 22, 24 | anbi12d 743 | . . 3 ⊢ (𝑓 = 𝑄 → ((𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑄 ∈ 𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))) |
26 | 15, 21, 25 | spcegf 3262 | . 2 ⊢ (𝑄 ∈ V → ((𝑄 ∈ 𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) → ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))) |
27 | 1, 26 | mpan9 485 | 1 ⊢ ((𝜒 ∧ 𝑄 ∈ V) → ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 [wsbc 3402 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 {csn 4125 〈cop 4131 ∪ cuni 4372 class class class wbr 4583 dom cdm 5038 ↾ cres 5040 Fn wfn 5799 ‘cfv 5804 predc-bnj14 30007 FrSe w-bnj15 30011 trClc-bnj18 30013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-dm 5048 df-res 5050 df-iota 5768 df-fv 5812 |
This theorem is referenced by: bnj1312 30380 |
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