Theorem bnj1423 35547

Description: Technical lemma for bnj60 35558. 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 264	. . . 4 ⊢ ((𝜒 ∧ 𝑧 ∈ 𝐸) ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))
18		biid 264	. . . 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 35545	. . 3 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ {𝑥}) → (𝑄‘𝑧) = (𝐺‘𝑊))
20		biid 264	. . . 4 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) ↔ ((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
21		biid 264	. . . 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 264	. . . 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 264	. . . 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 2762	. . . 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 35546	. . 3 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → (𝑄‘𝑧) = (𝐺‘𝑊))
26	14	bnj1424 35334	. . . 4 ⊢ (𝑧 ∈ 𝐸 → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
27	26	adantl 487	. . 3 ⊢ ((𝜒 ∧ 𝑧 ∈ 𝐸) → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
28	19, 25, 27	mpjaodan 973	. 2 ⊢ ((𝜒 ∧ 𝑧 ∈ 𝐸) → (𝑄‘𝑧) = (𝐺‘𝑊))
29	28	ralrimiva 3156	1 ⊢ (𝜒 → ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2740   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3414  [wsbc 3742   ∪ cun 3900   ⊆ wss 3902  ∅c0 4282  {csn 4587  ⟨cop 4593  ∪ cuni 4870   class class class wbr 5107  dom cdm 5659   ↾ cres 5661   Fn wfn 6532  ‘cfv 6537   ∧ w-bnj17 35183   predc-bnj14 35185   FrSe w-bnj15 35189   trClc-bnj18 35191

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-reg 9567  ax-inf2 9623

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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 7866  df-1o 8458  df-bnj17 35184  df-bnj14 35186  df-bnj13 35188  df-bnj15 35190  df-bnj18 35192

This theorem is used by:  bnj1312  35554