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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1371 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 33677. 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 |
---|---|
bnj1371.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1371.2 | ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ |
bnj1371.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1371.4 | ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
bnj1371.5 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} |
bnj1371.6 | ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) |
bnj1371.7 | ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) |
bnj1371.8 | ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) |
bnj1371.9 | ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} |
bnj1371.10 | ⊢ 𝑃 = ∪ 𝐻 |
bnj1371.11 | ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))) |
Ref | Expression |
---|---|
bnj1371 | ⊢ (𝑓 ∈ 𝐻 → Fun 𝑓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1371.9 | . . . . . . 7 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} | |
2 | 1 | bnj1436 33454 | . . . . . 6 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′) |
3 | rexex 3080 | . . . . . 6 ⊢ (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ → ∃𝑦𝜏′) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦𝜏′) |
5 | bnj1371.11 | . . . . . 6 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))) | |
6 | 5 | exbii 1851 | . . . . 5 ⊢ (∃𝑦𝜏′ ↔ ∃𝑦(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))) |
7 | 4, 6 | sylib 217 | . . . 4 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))) |
8 | exsimpl 1872 | . . . 4 ⊢ (∃𝑦(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∃𝑦 𝑓 ∈ 𝐶) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦 𝑓 ∈ 𝐶) |
10 | bnj1371.3 | . . . . . . 7 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
11 | 10 | eqabi 2882 | . . . . . 6 ⊢ (𝑓 ∈ 𝐶 ↔ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))) |
12 | 11 | bnj1238 33421 | . . . . 5 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑) |
13 | fnfun 6603 | . . . . 5 ⊢ (𝑓 Fn 𝑑 → Fun 𝑓) | |
14 | 12, 13 | bnj31 33334 | . . . 4 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 Fun 𝑓) |
15 | 14 | bnj1265 33427 | . . 3 ⊢ (𝑓 ∈ 𝐶 → Fun 𝑓) |
16 | 9, 15 | bnj593 33360 | . 2 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦Fun 𝑓) |
17 | 16 | bnj937 33386 | 1 ⊢ (𝑓 ∈ 𝐻 → Fun 𝑓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2714 ≠ wne 2944 ∀wral 3065 ∃wrex 3074 {crab 3408 [wsbc 3740 ∪ cun 3909 ⊆ wss 3911 ∅c0 4283 {csn 4587 ⟨cop 4593 ∪ cuni 4866 class class class wbr 5106 dom cdm 5634 ↾ cres 5636 Fun wfun 6491 Fn wfn 6492 ‘cfv 6497 predc-bnj14 33303 FrSe w-bnj15 33307 trClc-bnj18 33309 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rex 3075 df-fn 6500 |
This theorem is referenced by: bnj1384 33647 |
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