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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1416 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj60 35077. 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 5914 | . . 3 ⊢ dom 𝑄 = dom (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) | 
| 3 | dmun 5920 | . . 3 ⊢ dom (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) = (dom 𝑃 ∪ dom {〈𝑥, (𝐺‘𝑍)〉}) | |
| 4 | fvex 6918 | . . . . 5 ⊢ (𝐺‘𝑍) ∈ V | |
| 5 | 4 | dmsnop 6235 | . . . 4 ⊢ dom {〈𝑥, (𝐺‘𝑍)〉} = {𝑥} | 
| 6 | 5 | uneq2i 4164 | . . 3 ⊢ (dom 𝑃 ∪ dom {〈𝑥, (𝐺‘𝑍)〉}) = (dom 𝑃 ∪ {𝑥}) | 
| 7 | 2, 3, 6 | 3eqtri 2768 | . 2 ⊢ dom 𝑄 = (dom 𝑃 ∪ {𝑥}) | 
| 8 | bnj1416.28 | . . . 4 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) | |
| 9 | 8 | uneq1d 4166 | . . 3 ⊢ (𝜒 → (dom 𝑃 ∪ {𝑥}) = ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥})) | 
| 10 | uncom 4157 | . . 3 ⊢ ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) | |
| 11 | 9, 10 | eqtrdi 2792 | . 2 ⊢ (𝜒 → (dom 𝑃 ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) | 
| 12 | 7, 11 | eqtrid 2788 | 1 ⊢ (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∃wex 1778 ∈ wcel 2107 {cab 2713 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3435 [wsbc 3787 ∪ cun 3948 ⊆ wss 3950 ∅c0 4332 {csn 4625 〈cop 4631 ∪ cuni 4906 class class class wbr 5142 dom cdm 5684 ↾ cres 5686 Fn wfn 6555 ‘cfv 6560 predc-bnj14 34703 FrSe w-bnj15 34707 trClc-bnj18 34709 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-dm 5694 df-iota 6513 df-fv 6568 | 
| This theorem is referenced by: bnj1312 35073 | 
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