Theorem bnj1493 35392

Description: Technical lemma for bnj60 35395. 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 2769	. 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 2769	. 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 2769	. 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 2769	. 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 2769	. 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 2769	. 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 2769	. 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 35391	1 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  {crab 3423  [wsbc 3753   ∪ cun 3911   ⊆ wss 3913  ∅c0 4294  {csn 4594  ⟨cop 4600  ∪ cuni 4876   class class class wbr 5113  dom cdm 5662   ↾ cres 5664   Fn wfn 6532  ‘cfv 6537   predc-bnj14 35022   FrSe w-bnj15 35026   trClc-bnj18 35028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-reg 9554  ax-inf2 9610

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7863  df-1o 8453  df-bnj17 35021  df-bnj14 35023  df-bnj13 35025  df-bnj15 35027  df-bnj18 35029  df-bnj19 35031

This theorem is referenced by:  bnj1498  35394