Theorem bnj1493 35042

Description: Technical lemma for bnj60 35045. 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.)

Hypotheses

Ref	Expression
bnj1493.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1493.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1493.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}

Assertion

Ref	Expression
bnj1493	⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝐺,𝑑,𝑓,𝑥   𝑅,𝑑,𝑓,𝑥

Allowed substitution hints:   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝑌(𝑥,𝑓,𝑑)

Proof of Theorem bnj1493

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1493.1	. 2 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2		bnj1493.2	. 2 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
3		bnj1493.3	. 2 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
4		biid 261	. 2 ⊢ ((𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
5		eqid 2729	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))}
6		biid 261	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ≠ ∅) ↔ (𝑅 FrSe 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ≠ ∅))
7		biid 261	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ≠ ∅) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ¬ 𝑦𝑅𝑥) ↔ ((𝑅 FrSe 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ≠ ∅) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ¬ 𝑦𝑅𝑥))
8		biid 261	. 2 ⊢ ([𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ [𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
9		eqid 2729	. 2 ⊢ {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)[𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)[𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))}
10		eqid 2729	. 2 ⊢ ∪ {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)[𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} = ∪ {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)[𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))}
11		eqid 2729	. 2 ⊢ ⟨𝑥, (∪ {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)[𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (∪ {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)[𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ↾ pred(𝑥, 𝐴, 𝑅))⟩
12		eqid 2729	. 2 ⊢ (∪ {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)[𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ∪ {⟨𝑥, (𝐺‘⟨𝑥, (∪ {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)[𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ↾ pred(𝑥, 𝐴, 𝑅))⟩)⟩}) = (∪ {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)[𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ∪ {⟨𝑥, (𝐺‘⟨𝑥, (∪ {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)[𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ↾ pred(𝑥, 𝐴, 𝑅))⟩)⟩})
13		eqid 2729	. 2 ⊢ ⟨𝑧, ((∪ {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)[𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ∪ {⟨𝑥, (𝐺‘⟨𝑥, (∪ {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)[𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ↾ pred(𝑥, 𝐴, 𝑅))⟩)⟩}) ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑧, ((∪ {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)[𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ∪ {⟨𝑥, (𝐺‘⟨𝑥, (∪ {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)[𝑦 / 𝑥](𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))} ↾ pred(𝑥, 𝐴, 𝑅))⟩)⟩}) ↾ pred(𝑧, 𝐴, 𝑅))⟩
14		eqid 2729	. 2 ⊢ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	bnj1312 35041	1 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3402  [wsbc 3750   ∪ cun 3909   ⊆ wss 3911  ∅c0 4292  {csn 4585  ⟨cop 4591  ∪ cuni 4867   class class class wbr 5102  dom cdm 5631   ↾ cres 5633   Fn wfn 6494  ‘cfv 6499   predc-bnj14 34671   FrSe w-bnj15 34675   trClc-bnj18 34677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-reg 9521  ax-inf2 9570

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-1o 8411  df-bnj17 34670  df-bnj14 34672  df-bnj13 34674  df-bnj15 34676  df-bnj18 34678  df-bnj19 34680

This theorem is referenced by:  bnj1498  35044