Theorem bnj1442 30177

Description: Technical lemma for bnj60 30190. 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
bnj1442.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1442.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1442.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1442.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1442.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1442.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1442.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1442.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1442.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1442.10	⊢ 𝑃 = ∪ 𝐻
bnj1442.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1442.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
bnj1442.13	⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1442.14	⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
bnj1442.15	⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
bnj1442.16	⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
bnj1442.17	⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))
bnj1442.18	⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥}))

Assertion

Ref	Expression
bnj1442	⊢ (𝜂 → (𝑄‘𝑧) = (𝐺‘𝑊))

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem bnj1442

Step	Hyp	Ref	Expression
1		bnj1442.18	. . 3 ⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥}))
2		bnj1442.17	. . . 4 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))
3		bnj1442.16	. . . . . 6 ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
4	3	bnj930 29900	. . . . 5 ⊢ (𝜒 → Fun 𝑄)
5		opex 4853	. . . . . . . 8 ⊢ ⟨𝑥, (𝐺‘𝑍)⟩ ∈ V
6	5	snid 4154	. . . . . . 7 ⊢ ⟨𝑥, (𝐺‘𝑍)⟩ ∈ {⟨𝑥, (𝐺‘𝑍)⟩}
7		elun2 3742	. . . . . . 7 ⊢ (⟨𝑥, (𝐺‘𝑍)⟩ ∈ {⟨𝑥, (𝐺‘𝑍)⟩} → ⟨𝑥, (𝐺‘𝑍)⟩ ∈ (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}))
8	6, 7	ax-mp 5	. . . . . 6 ⊢ ⟨𝑥, (𝐺‘𝑍)⟩ ∈ (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
9		bnj1442.12	. . . . . 6 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
10	8, 9	eleqtrri 2686	. . . . 5 ⊢ ⟨𝑥, (𝐺‘𝑍)⟩ ∈ 𝑄
11		funopfv 6130	. . . . 5 ⊢ (Fun 𝑄 → (⟨𝑥, (𝐺‘𝑍)⟩ ∈ 𝑄 → (𝑄‘𝑥) = (𝐺‘𝑍)))
12	4, 10, 11	mpisyl 21	. . . 4 ⊢ (𝜒 → (𝑄‘𝑥) = (𝐺‘𝑍))
13	2, 12	bnj832 29888	. . 3 ⊢ (𝜃 → (𝑄‘𝑥) = (𝐺‘𝑍))
14	1, 13	bnj832 29888	. 2 ⊢ (𝜂 → (𝑄‘𝑥) = (𝐺‘𝑍))
15		elsni 4141	. . . 4 ⊢ (𝑧 ∈ {𝑥} → 𝑧 = 𝑥)
16	1, 15	simplbiim 656	. . 3 ⊢ (𝜂 → 𝑧 = 𝑥)
17	16	fveq2d 6092	. 2 ⊢ (𝜂 → (𝑄‘𝑧) = (𝑄‘𝑥))
18		bnj602 30045	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅))
19	18	reseq2d 5304	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)))
20	16, 19	syl 17	. . . . . 6 ⊢ (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)))
21	9	bnj931 29901	. . . . . . . . . 10 ⊢ 𝑃 ⊆ 𝑄
22	21	a1i 11	. . . . . . . . 9 ⊢ (𝜒 → 𝑃 ⊆ 𝑄)
23		bnj1442.7	. . . . . . . . . . . 12 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
24		bnj1442.6	. . . . . . . . . . . . 13 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
25	24	simplbi 474	. . . . . . . . . . . 12 ⊢ (𝜓 → 𝑅 FrSe 𝐴)
26	23, 25	bnj835 29889	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑅 FrSe 𝐴)
27		bnj1442.5	. . . . . . . . . . . 12 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
28	27, 23	bnj1212 29930	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑥 ∈ 𝐴)
29		bnj906 30060	. . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
30	26, 28, 29	syl2anc 690	. . . . . . . . . 10 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
31		bnj1442.15	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
32		fndm 5890	. . . . . . . . . . 11 ⊢ (𝑃 Fn trCl(𝑥, 𝐴, 𝑅) → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
33	31, 32	syl 17	. . . . . . . . . 10 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
34	30, 33	sseqtr4d 3604	. . . . . . . . 9 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑃)
35	4, 22, 34	bnj1503 29979	. . . . . . . 8 ⊢ (𝜒 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
36	2, 35	bnj832 29888	. . . . . . 7 ⊢ (𝜃 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
37	1, 36	bnj832 29888	. . . . . 6 ⊢ (𝜂 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
38	20, 37	eqtrd 2643	. . . . 5 ⊢ (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
39	16, 38	opeq12d 4342	. . . 4 ⊢ (𝜂 → ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
40		bnj1442.13	. . . 4 ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
41		bnj1442.11	. . . 4 ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
42	39, 40, 41	3eqtr4g 2668	. . 3 ⊢ (𝜂 → 𝑊 = 𝑍)
43	42	fveq2d 6092	. 2 ⊢ (𝜂 → (𝐺‘𝑊) = (𝐺‘𝑍))
44	14, 17, 43	3eqtr4d 2653	1 ⊢ (𝜂 → (𝑄‘𝑧) = (𝐺‘𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 194   ∧ wa 382   ∧ w3a 1030   = wceq 1474  ∃wex 1694   ∈ wcel 1976  {cab 2595   ≠ wne 2779  ∀wral 2895  ∃wrex 2896  {crab 2899  [wsbc 3401   ∪ cun 3537   ⊆ wss 3539  ∅c0 3873  {csn 4124  ⟨cop 4130  ∪ cuni 4366   class class class wbr 4577  dom cdm 5028   ↾ cres 5030  Fun wfun 5784   Fn wfn 5785  ‘cfv 5790   predc-bnj14 29813   FrSe w-bnj15 29817   trClc-bnj18 29819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-reg 8357  ax-inf2 8398

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-om 6935  df-1o 7424  df-bnj17 29812  df-bnj14 29814  df-bnj13 29816  df-bnj15 29818  df-bnj18 29820

This theorem is referenced by:  bnj1423  30179