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| 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 | 
|---|---|
| bnj1374.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} | 
| bnj1374.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 | 
| bnj1374.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | 
| bnj1374.4 | ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) | 
| bnj1374.5 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} | 
| bnj1374.6 | ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) | 
| bnj1374.7 | ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) | 
| bnj1374.8 | ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) | 
| bnj1374.9 | ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} | 
| Ref | Expression | 
|---|---|
| bnj1374 | ⊢ (𝑓 ∈ 𝐻 → 𝑓 ∈ 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bnj1374.9 | . . . . . 6 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} | |
| 2 | 1 | bnj1436 34854 | . . . . 5 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′) | 
| 3 | rexex 3075 | . . . . 5 ⊢ (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ → ∃𝑦𝜏′) | |
| 4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦𝜏′) | 
| 5 | bnj1374.1 | . . . . . 6 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} | |
| 6 | bnj1374.2 | . . . . . 6 ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
| 7 | bnj1374.3 | . . . . . 6 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
| 8 | bnj1374.4 | . . . . . 6 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) | |
| 9 | bnj1374.8 | . . . . . 6 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) | |
| 10 | 5, 6, 7, 8, 9 | bnj1373 35045 | . . . . 5 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))) | 
| 11 | 10 | exbii 1847 | . . . 4 ⊢ (∃𝑦𝜏′ ↔ ∃𝑦(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))) | 
| 12 | 4, 11 | sylib 218 | . . 3 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))) | 
| 13 | exsimpl 1867 | . . 3 ⊢ (∃𝑦(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∃𝑦 𝑓 ∈ 𝐶) | |
| 14 | 12, 13 | syl 17 | . 2 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦 𝑓 ∈ 𝐶) | 
| 15 | 14 | bnj937 34786 | 1 ⊢ (𝑓 ∈ 𝐻 → 𝑓 ∈ 𝐶) | 
| 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 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-11 2156 ax-12 2176 ax-ext 2707 | 
| 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-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 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-iun 4992 df-br 5143 df-bnj14 34704 df-bnj18 34710 | 
| This theorem is referenced by: bnj1384 35047 | 
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