Theorem bnj1423 35034

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
bnj1423.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1423.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1423.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1423.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1423.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1423.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1423.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1423.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1423.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1423.10	⊢ 𝑃 = ∪ 𝐻
bnj1423.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1423.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
bnj1423.13	⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1423.14	⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
bnj1423.15	⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
bnj1423.16	⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

Assertion

Ref	Expression
bnj1423	⊢ (𝜒 → ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊))

Distinct variable groups:   𝐴,𝑑,𝑓,𝑥,𝑦,𝑧   𝐵,𝑓   𝑦,𝐷   𝐸,𝑑,𝑓,𝑦   𝐺,𝑑,𝑓,𝑥,𝑦,𝑧   𝑅,𝑑,𝑓,𝑥,𝑦,𝑧   𝑧,𝑌   𝜒,𝑧   𝜓,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑧,𝑑)   𝐶(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐸(𝑥,𝑧)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑊(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑧,𝑓,𝑑)

Proof of Theorem bnj1423

Step	Hyp	Ref	Expression
1		bnj1423.1	. . . 4 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2		bnj1423.2	. . . 4 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
3		bnj1423.3	. . . 4 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
4		bnj1423.4	. . . 4 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
5		bnj1423.5	. . . 4 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
6		bnj1423.6	. . . 4 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
7		bnj1423.7	. . . 4 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
8		bnj1423.8	. . . 4 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
9		bnj1423.9	. . . 4 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
10		bnj1423.10	. . . 4 ⊢ 𝑃 = ∪ 𝐻
11		bnj1423.11	. . . 4 ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
12		bnj1423.12	. . . 4 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
13		bnj1423.13	. . . 4 ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
14		bnj1423.14	. . . 4 ⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
15		bnj1423.15	. . . 4 ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
16		bnj1423.16	. . . 4 ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
17		biid 261	. . . 4 ⊢ ((𝜒 ∧ 𝑧 ∈ 𝐸) ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))
18		biid 261	. . . 4 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ {𝑥}) ↔ ((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ {𝑥}))
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	bnj1442 35032	. . 3 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ {𝑥}) → (𝑄‘𝑧) = (𝐺‘𝑊))
20		biid 261	. . . 4 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) ↔ ((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
21		biid 261	. . . 4 ⊢ ((((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓) ↔ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))
22		biid 261	. . . 4 ⊢ (((((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓) ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ↔ ((((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓) ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
23		biid 261	. . . 4 ⊢ ((((((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓) ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ∧ 𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ (((((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓) ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ∧ 𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
24		eqid 2729	. . . 4 ⊢ ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩
25	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24	bnj1450 35033	. . 3 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → (𝑄‘𝑧) = (𝐺‘𝑊))
26	14	bnj1424 34821	. . . 4 ⊢ (𝑧 ∈ 𝐸 → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
27	26	adantl 481	. . 3 ⊢ ((𝜒 ∧ 𝑧 ∈ 𝐸) → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
28	19, 25, 27	mpjaodan 960	. 2 ⊢ ((𝜒 ∧ 𝑧 ∈ 𝐸) → (𝑄‘𝑧) = (𝐺‘𝑊))
29	28	ralrimiva 3121	1 ⊢ (𝜒 → ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3394  [wsbc 3742   ∪ cun 3901   ⊆ wss 3903  ∅c0 4284  {csn 4577  ⟨cop 4583  ∪ cuni 4858   class class class wbr 5092  dom cdm 5619   ↾ cres 5621   Fn wfn 6477  ‘cfv 6482   ∧ w-bnj17 34669   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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-reg 9484  ax-inf2 9537

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 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-om 7800  df-1o 8388  df-bnj17 34670  df-bnj14 34672  df-bnj13 34674  df-bnj15 34676  df-bnj18 34678

This theorem is referenced by:  bnj1312  35041