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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1416 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 33028. 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 |
---|---|
bnj1416.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1416.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1416.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1416.4 | ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
bnj1416.5 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} |
bnj1416.6 | ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) |
bnj1416.7 | ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) |
bnj1416.8 | ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) |
bnj1416.9 | ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} |
bnj1416.10 | ⊢ 𝑃 = ∪ 𝐻 |
bnj1416.11 | ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1416.12 | ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) |
bnj1416.28 | ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) |
Ref | Expression |
---|---|
bnj1416 | ⊢ (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1416.12 | . . . 4 ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) | |
2 | 1 | dmeqi 5807 | . . 3 ⊢ dom 𝑄 = dom (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) |
3 | dmun 5813 | . . 3 ⊢ dom (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) = (dom 𝑃 ∪ dom {〈𝑥, (𝐺‘𝑍)〉}) | |
4 | fvex 6780 | . . . . 5 ⊢ (𝐺‘𝑍) ∈ V | |
5 | 4 | dmsnop 6113 | . . . 4 ⊢ dom {〈𝑥, (𝐺‘𝑍)〉} = {𝑥} |
6 | 5 | uneq2i 4094 | . . 3 ⊢ (dom 𝑃 ∪ dom {〈𝑥, (𝐺‘𝑍)〉}) = (dom 𝑃 ∪ {𝑥}) |
7 | 2, 3, 6 | 3eqtri 2770 | . 2 ⊢ dom 𝑄 = (dom 𝑃 ∪ {𝑥}) |
8 | bnj1416.28 | . . . 4 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) | |
9 | 8 | uneq1d 4096 | . . 3 ⊢ (𝜒 → (dom 𝑃 ∪ {𝑥}) = ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥})) |
10 | uncom 4087 | . . 3 ⊢ ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) | |
11 | 9, 10 | eqtrdi 2794 | . 2 ⊢ (𝜒 → (dom 𝑃 ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) |
12 | 7, 11 | eqtrid 2790 | 1 ⊢ (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∃wex 1782 ∈ wcel 2106 {cab 2715 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 {crab 3068 [wsbc 3716 ∪ cun 3885 ⊆ wss 3887 ∅c0 4257 {csn 4562 〈cop 4568 ∪ cuni 4840 class class class wbr 5074 dom cdm 5585 ↾ cres 5587 Fn wfn 6422 ‘cfv 6427 predc-bnj14 32653 FrSe w-bnj15 32657 trClc-bnj18 32659 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5222 ax-nul 5229 ax-pr 5351 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-br 5075 df-dm 5595 df-iota 6385 df-fv 6435 |
This theorem is referenced by: bnj1312 33024 |
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