Theorem bnj1321 35424

Description: Technical lemma for bnj60 35459. 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
bnj1321.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1321.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1321.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1321.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))

Assertion

Ref	Expression
bnj1321	⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃!𝑓𝜏)

Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝐺,𝑑,𝑓   𝑅,𝑑,𝑓,𝑥

Allowed substitution hints:   𝜏(𝑥, 𝑓, 𝑑)   𝐵(𝑥, 𝑑)   𝐶(𝑥, 𝑓, 𝑑)   𝐺(𝑥)   𝑌(𝑥, 𝑓, 𝑑)

Proof of Theorem bnj1321

Dummy variables 𝑔 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 489	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃𝑓𝜏)
2		simp1 1153	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑅 FrSe 𝐴)
3		bnj1321.4	. . . . . . . . 9 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4	3	simplbi 501	. . . . . . . 8 ⊢ (𝜏 → 𝑓 ∈ 𝐶)
5	4	3ad2ant2 1151	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 ∈ 𝐶)
6		bnj1321.3	. . . . . . . . . . . . 13 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
7		nfab1 2926	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑓{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
8	6, 7	nfcxfr 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓𝐶
9	8	nfcri 2916	. . . . . . . . . . 11 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐶
10		nfv 1943	. . . . . . . . . . 11 ⊢ Ⅎ𝑓dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
11	9, 10	nfan 1928	. . . . . . . . . 10 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
12		eleq1w 2845	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶))
13		dmeq 5892	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
14	13	eqeq1d 2764	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
15	12, 14	anbi12d 643	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑔 ∈ 𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))))
16	3, 15	bitrid 286	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝜏 ↔ (𝑔 ∈ 𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))))
17	11, 16	sbiev 2346	. . . . . . . . 9 ⊢ ([𝑔 / 𝑓]𝜏 ↔ (𝑔 ∈ 𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
18	17	simplbi 501	. . . . . . . 8 ⊢ ([𝑔 / 𝑓]𝜏 → 𝑔 ∈ 𝐶)
19	18	3ad2ant3 1152	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑔 ∈ 𝐶)
20		bnj1321.1	. . . . . . . 8 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
21		bnj1321.2	. . . . . . . 8 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
22		eqid 2762	. . . . . . . 8 ⊢ (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
23	20, 21, 6, 22	bnj1326 35423	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐶 ∧ 𝑔 ∈ 𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
24	2, 5, 19, 23	syl3anc 1397	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
25	3	simprbi 502	. . . . . . . . . 10 ⊢ (𝜏 → dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
26	25	3ad2ant2 1151	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
27	17	simprbi 502	. . . . . . . . . 10 ⊢ ([𝑔 / 𝑓]𝜏 → dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
28	27	3ad2ant3 1152	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
29	26, 28	eqtr4d 2800	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑓 = dom 𝑔)
30		bnj1322 35219	. . . . . . . . 9 ⊢ (dom 𝑓 = dom 𝑔 → (dom 𝑓 ∩ dom 𝑔) = dom 𝑓)
31	30	reseq2d 5977	. . . . . . . 8 ⊢ (dom 𝑓 = dom 𝑔 → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑓 ↾ dom 𝑓))
32	29, 31	syl 18	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑓 ↾ dom 𝑓))
33		releq 5762	. . . . . . . . 9 ⊢ (𝑧 = 𝑓 → (Rel 𝑧 ↔ Rel 𝑓))
34	20, 21, 6	bnj66 35257	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐶 → Rel 𝑧)
35	33, 34	vtoclga 3540	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐶 → Rel 𝑓)
36		resdm 6024	. . . . . . . 8 ⊢ (Rel 𝑓 → (𝑓 ↾ dom 𝑓) = 𝑓)
37	5, 35, 36	3syl 19	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ dom 𝑓) = 𝑓)
38	32, 37	eqtrd 2797	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = 𝑓)
39		eqeq2 2774	. . . . . . . . . 10 ⊢ (dom 𝑓 = dom 𝑔 → ((dom 𝑓 ∩ dom 𝑔) = dom 𝑓 ↔ (dom 𝑓 ∩ dom 𝑔) = dom 𝑔))
40	30, 39	mpbid 235	. . . . . . . . 9 ⊢ (dom 𝑓 = dom 𝑔 → (dom 𝑓 ∩ dom 𝑔) = dom 𝑔)
41	40	reseq2d 5977	. . . . . . . 8 ⊢ (dom 𝑓 = dom 𝑔 → (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ dom 𝑔))
42	29, 41	syl 18	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ dom 𝑔))
43	20, 21, 6	bnj66 35257	. . . . . . . 8 ⊢ (𝑔 ∈ 𝐶 → Rel 𝑔)
44		resdm 6024	. . . . . . . 8 ⊢ (Rel 𝑔 → (𝑔 ↾ dom 𝑔) = 𝑔)
45	19, 43, 44	3syl 19	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ dom 𝑔) = 𝑔)
46	42, 45	eqtrd 2797	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)) = 𝑔)
47	24, 38, 46	3eqtr3d 2805	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔)
48	47	3expib 1139	. . . 4 ⊢ (𝑅 FrSe 𝐴 → ((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
49	48	alrimivv 1957	. . 3 ⊢ (𝑅 FrSe 𝐴 → ∀𝑓∀𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
50	49	adantr 485	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∀𝑓∀𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
51		nfv 1943	. . 3 ⊢ Ⅎ𝑔𝜏
52	51	eu2 2636	. 2 ⊢ (∃!𝑓𝜏 ↔ (∃𝑓𝜏 ∧ ∀𝑓∀𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔)))
53	1, 50, 52	sylanbrc 594	1 ⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃!𝑓𝜏)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102  ∀wal 1567   = wceq 1569  ∃wex 1808  [wsb 2095   ∈ wcel 2142  ∃!weu 2595  {cab 2740  ∀wral 3078  ∃wrex 3088   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  {csn 4588  ⟨cop 4594  dom cdm 5660   ↾ cres 5662  Rel wrel 5665   Fn wfn 6531  ‘cfv 6536   predc-bnj14 35086   FrSe w-bnj15 35090   trClc-bnj18 35092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-reg 9552  ax-inf2 9608

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7861  df-1o 8451  df-bnj17 35085  df-bnj14 35087  df-bnj13 35089  df-bnj15 35091  df-bnj18 35093  df-bnj19 35095

This theorem is used by:  bnj1489  35453