Theorem bnj1493 35215

Description: Technical lemma for bnj60 35218. 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 2736	. 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 2736	. 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 2736	. 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 2736	. 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 2736	. 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 2736	. 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 2736	. 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 35214	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 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3399  [wsbc 3740   ∪ cun 3899   ⊆ wss 3901  ∅c0 4285  {csn 4580  ⟨cop 4586  ∪ cuni 4863   class class class wbr 5098  dom cdm 5624   ↾ cres 5626   Fn wfn 6487  ‘cfv 6492   predc-bnj14 34844   FrSe w-bnj15 34848   trClc-bnj18 34850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-reg 9497  ax-inf2 9550

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-1o 8397  df-bnj17 34843  df-bnj14 34845  df-bnj13 34847  df-bnj15 34849  df-bnj18 34851  df-bnj19 34853

This theorem is referenced by:  bnj1498  35217