Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1416 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 32329. 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 5767 | . . 3 ⊢ dom 𝑄 = dom (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) |
3 | dmun 5773 | . . 3 ⊢ dom (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) = (dom 𝑃 ∪ dom {〈𝑥, (𝐺‘𝑍)〉}) | |
4 | fvex 6677 | . . . . 5 ⊢ (𝐺‘𝑍) ∈ V | |
5 | 4 | dmsnop 6067 | . . . 4 ⊢ dom {〈𝑥, (𝐺‘𝑍)〉} = {𝑥} |
6 | 5 | uneq2i 4135 | . . 3 ⊢ (dom 𝑃 ∪ dom {〈𝑥, (𝐺‘𝑍)〉}) = (dom 𝑃 ∪ {𝑥}) |
7 | 2, 3, 6 | 3eqtri 2848 | . 2 ⊢ dom 𝑄 = (dom 𝑃 ∪ {𝑥}) |
8 | bnj1416.28 | . . . 4 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) | |
9 | 8 | uneq1d 4137 | . . 3 ⊢ (𝜒 → (dom 𝑃 ∪ {𝑥}) = ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥})) |
10 | uncom 4128 | . . 3 ⊢ ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) | |
11 | 9, 10 | syl6eq 2872 | . 2 ⊢ (𝜒 → (dom 𝑃 ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) |
12 | 7, 11 | syl5eq 2868 | 1 ⊢ (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 [wsbc 3771 ∪ cun 3933 ⊆ wss 3935 ∅c0 4290 {csn 4560 〈cop 4566 ∪ cuni 4831 class class class wbr 5058 dom cdm 5549 ↾ cres 5551 Fn wfn 6344 ‘cfv 6349 predc-bnj14 31953 FrSe w-bnj15 31957 trClc-bnj18 31959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-dm 5559 df-iota 6308 df-fv 6357 |
This theorem is referenced by: bnj1312 32325 |
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