Theorem bnj1321 35657

Description: Technical lemma for bnj60 35692. 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 490	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃𝑓𝜏)
2		simp1 1154	. . . . . . 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 502	. . . . . . . 8 ⊢ (𝜏 → 𝑓 ∈ 𝐶)
5	4	3ad2ant2 1152	. . . . . . 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 2924	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑓{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
8	6, 7	nfcxfr 2920	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓𝐶
9	8	nfcri 2914	. . . . . . . . . . 11 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐶
10		nfv 1946	. . . . . . . . . . 11 ⊢ Ⅎ𝑓dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
11	9, 10	nfan 1931	. . . . . . . . . 10 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
12		eleq1w 2843	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶))
13		dmeq 5884	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
14	13	eqeq1d 2762	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
15	12, 14	anbi12d 644	. . . . . . . . . . 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 2345	. . . . . . . . 9 ⊢ ([𝑔 / 𝑓]𝜏 ↔ (𝑔 ∈ 𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
18	17	simplbi 502	. . . . . . . 8 ⊢ ([𝑔 / 𝑓]𝜏 → 𝑔 ∈ 𝐶)
19	18	3ad2ant3 1153	. . . . . . 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 2760	. . . . . . . 8 ⊢ (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
23	20, 21, 6, 22	bnj1326 35656	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐶 ∧ 𝑔 ∈ 𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
24	2, 5, 19, 23	syl3anc 1398	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
25	3	simprbi 503	. . . . . . . . . 10 ⊢ (𝜏 → dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
26	25	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
27	17	simprbi 503	. . . . . . . . . 10 ⊢ ([𝑔 / 𝑓]𝜏 → dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
28	27	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
29	26, 28	eqtr4d 2798	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑓 = dom 𝑔)
30		bnj1322 35452	. . . . . . . . 9 ⊢ (dom 𝑓 = dom 𝑔 → (dom 𝑓 ∩ dom 𝑔) = dom 𝑓)
31	30	reseq2d 5969	. . . . . . . 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 5752	. . . . . . . . 9 ⊢ (𝑧 = 𝑓 → (Rel 𝑧 ↔ Rel 𝑓))
34	20, 21, 6	bnj66 35490	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐶 → Rel 𝑧)
35	33, 34	vtoclga 3536	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐶 → Rel 𝑓)
36		resdm 6014	. . . . . . . 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 2795	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = 𝑓)
39		eqeq2 2772	. . . . . . . . . 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 5969	. . . . . . . 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 35490	. . . . . . . 8 ⊢ (𝑔 ∈ 𝐶 → Rel 𝑔)
44		resdm 6014	. . . . . . . 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 2795	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)) = 𝑔)
47	24, 38, 46	3eqtr3d 2803	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔)
48	47	3expib 1140	. . . 4 ⊢ (𝑅 FrSe 𝐴 → ((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
49	48	alrimivv 1960	. . 3 ⊢ (𝑅 FrSe 𝐴 → ∀𝑓∀𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
50	49	adantr 486	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∀𝑓∀𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
51		nfv 1946	. . 3 ⊢ Ⅎ𝑔𝜏
52	51	eu2 2634	. 2 ⊢ (∃!𝑓𝜏 ↔ (∃𝑓𝜏 ∧ ∀𝑓∀𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔)))
53	1, 50, 52	sylanbrc 595	1 ⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃!𝑓𝜏)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569  ∃wex 1811  [wsb 2098   ∈ wcel 2144  ∃!weu 2593  {cab 2738  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ∩ cin 4089   ∪ cun 4149  {csn 4583  ⟨cop 4589  dom cdm 5650   ↾ cres 5652  Rel wrel 5655   Fn wfn 6533  ‘cfv 6538   predc-bnj14 35319   FrSe w-bnj15 35323   trClc-bnj18 35325

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-reg 9586  ax-inf2 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-1o 8476  df-bnj17 35318  df-bnj14 35320  df-bnj13 35322  df-bnj15 35324  df-bnj18 35326  df-bnj19 35328

This theorem is used by:  bnj1489  35686