Theorem bnj1312 32609

Description: Technical lemma for bnj60 32613. 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
bnj1312.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1312.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1312.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1312.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1312.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1312.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1312.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1312.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1312.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1312.10	⊢ 𝑃 = ∪ 𝐻
bnj1312.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1312.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
bnj1312.13	⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1312.14	⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))

Assertion

Ref	Expression
bnj1312	⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

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

Allowed substitution hints:   𝜓(𝑥,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑧,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑧,𝑑)   𝐶(𝑥,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑓,𝑑)   𝐸(𝑥)   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 bnj1312

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1312.5	. . 3 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
2		bnj1312.6	. . . 4 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
3	2	simplbi 501	. . . . . . 7 ⊢ (𝜓 → 𝑅 FrSe 𝐴)
4	1	ssrab3 3971	. . . . . . . 8 ⊢ 𝐷 ⊆ 𝐴
5	4	a1i 11	. . . . . . 7 ⊢ (𝜓 → 𝐷 ⊆ 𝐴)
6	2	simprbi 500	. . . . . . 7 ⊢ (𝜓 → 𝐷 ≠ ∅)
7	1	bnj1230 32353	. . . . . . . 8 ⊢ (𝑤 ∈ 𝐷 → ∀𝑥 𝑤 ∈ 𝐷)
8	7	bnj1228 32562	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ 𝐷 ≠ ∅) → ∃𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
9	3, 5, 6, 8	syl3anc 1372	. . . . . 6 ⊢ (𝜓 → ∃𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
10		bnj1312.7	. . . . . 6 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
11		nfv 1921	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑅 FrSe 𝐴
12	7	nfcii 2883	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐷
13		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑥∅
14	12, 13	nfne 3034	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐷 ≠ ∅
15	11, 14	nfan 1906	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)
16	2, 15	nfxfr 1859	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
17	16	nf5ri 2197	. . . . . 6 ⊢ (𝜓 → ∀𝑥𝜓)
18	9, 10, 17	bnj1521 32402	. . . . 5 ⊢ (𝜓 → ∃𝑥𝜒)
19	10	simp2bi 1147	. . . . 5 ⊢ (𝜒 → 𝑥 ∈ 𝐷)
20	1	bnj1538 32406	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → ¬ ∃𝑓𝜏)
21		bnj1312.1	. . . . . . . . 9 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
22		bnj1312.2	. . . . . . . . 9 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
23		bnj1312.3	. . . . . . . . 9 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
24		bnj1312.4	. . . . . . . . 9 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
25		bnj1312.8	. . . . . . . . 9 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
26		bnj1312.9	. . . . . . . . 9 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
27		bnj1312.10	. . . . . . . . 9 ⊢ 𝑃 = ∪ 𝐻
28		bnj1312.11	. . . . . . . . 9 ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
29		bnj1312.12	. . . . . . . . 9 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
30	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29	bnj1489 32607	. . . . . . . 8 ⊢ (𝜒 → 𝑄 ∈ V)
31		bnj1312.13	. . . . . . . . . . 11 ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
32		bnj1312.14	. . . . . . . . . . 11 ⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
33	10, 3	bnj835 32309	. . . . . . . . . . . . . 14 ⊢ (𝜒 → 𝑅 FrSe 𝐴)
34	21, 22, 23, 24, 1, 2, 10, 25, 26, 27	bnj1384 32583	. . . . . . . . . . . . . 14 ⊢ (𝑅 FrSe 𝐴 → Fun 𝑃)
35	33, 34	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜒 → Fun 𝑃)
36	21, 22, 23, 24, 1, 2, 10, 25, 26, 27	bnj1415 32589	. . . . . . . . . . . . 13 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
37	35, 36	bnj1422 32388	. . . . . . . . . . . 12 ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
38	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 36	bnj1416 32590	. . . . . . . . . . . . . 14 ⊢ (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
39	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 35, 38, 36	bnj1421 32593	. . . . . . . . . . . . 13 ⊢ (𝜒 → Fun 𝑄)
40	39, 38	bnj1422 32388	. . . . . . . . . . . 12 ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
41	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 37, 40	bnj1423 32602	. . . . . . . . . . 11 ⊢ (𝜒 → ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊))
42	32	fneq2i 6436	. . . . . . . . . . . 12 ⊢ (𝑄 Fn 𝐸 ↔ 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
43	40, 42	sylibr 237	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑄 Fn 𝐸)
44	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32	bnj1452 32603	. . . . . . . . . . 11 ⊢ (𝜒 → 𝐸 ∈ 𝐵)
45	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 30, 41, 43, 44	bnj1463 32606	. . . . . . . . . 10 ⊢ (𝜒 → 𝑄 ∈ 𝐶)
46	45, 38	jca 515	. . . . . . . . 9 ⊢ (𝜒 → (𝑄 ∈ 𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
47	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 46	bnj1491 32608	. . . . . . . 8 ⊢ ((𝜒 ∧ 𝑄 ∈ V) → ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
48	30, 47	mpdan 687	. . . . . . 7 ⊢ (𝜒 → ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
49	48, 24	bnj1198 32346	. . . . . 6 ⊢ (𝜒 → ∃𝑓𝜏)
50	20, 49	nsyl3 140	. . . . 5 ⊢ (𝜒 → ¬ 𝑥 ∈ 𝐷)
51	18, 19, 50	bnj1304 32370	. . . 4 ⊢ ¬ 𝜓
52	2, 51	bnj1541 32407	. . 3 ⊢ (𝑅 FrSe 𝐴 → 𝐷 = ∅)
53	1, 52	bnj1476 32398	. 2 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓𝜏)
54	24	exbii 1854	. . . 4 ⊢ (∃𝑓𝜏 ↔ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
55		df-rex 3059	. . . 4 ⊢ (∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
56	54, 55	bitr4i 281	. . 3 ⊢ (∃𝑓𝜏 ↔ ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
57	56	ralbii 3080	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑓𝜏 ↔ ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
58	53, 57	sylib 221	1 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542  ∃wex 1786   ∈ wcel 2114  {cab 2716   ≠ wne 2934  ∀wral 3053  ∃wrex 3054  {crab 3057  Vcvv 3398  [wsbc 3680   ∪ cun 3841   ⊆ wss 3843  ∅c0 4211  {csn 4516  ⟨cop 4522  ∪ cuni 4796   class class class wbr 5030  dom cdm 5525   ↾ cres 5527  Fun wfun 6333   Fn wfn 6334  ‘cfv 6339   predc-bnj14 32237   FrSe w-bnj15 32241   trClc-bnj18 32243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-reg 9129  ax-inf2 9177

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7600  df-1o 8131  df-bnj17 32236  df-bnj14 32238  df-bnj13 32240  df-bnj15 32242  df-bnj18 32244  df-bnj19 32246

This theorem is referenced by:  bnj1493  32610