Theorem bnj1493 35512

Description: Technical lemma for bnj60 35515. 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 264	. 2 ⊢ ((𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
5		eqid 2765	. 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 264	. 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 264	. 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 264	. 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 2765	. 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 2765	. 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 2765	. 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 2765	. 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 2765	. 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 2765	. 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 35511	1 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  {crab 3418  [wsbc 3746   ∪ cun 3904   ⊆ wss 3906  ∅c0 4286  {csn 4591  ⟨cop 4597  ∪ cuni 4874   class class class wbr 5111  dom cdm 5663   ↾ cres 5665   Fn wfn 6535  ‘cfv 6540   predc-bnj14 35142   FrSe w-bnj15 35146   trClc-bnj18 35148

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-reg 9561  ax-inf2 9617

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7869  df-1o 8459  df-bnj17 35141  df-bnj14 35143  df-bnj13 35145  df-bnj15 35147  df-bnj18 35149  df-bnj19 35151

This theorem is used by:  bnj1498  35514