Theorem bnj1321 34038

Description: Technical lemma for bnj60 34073. 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 486	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃𝑓𝜏)
2		simp1 1137	. . . . . . 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 499	. . . . . . . 8 ⊢ (𝜏 → 𝑓 ∈ 𝐶)
5	4	3ad2ant2 1135	. . . . . . 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 2906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑓{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
8	6, 7	nfcxfr 2902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓𝐶
9	8	nfcri 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐶
10		nfv 1918	. . . . . . . . . . 11 ⊢ Ⅎ𝑓dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
11	9, 10	nfan 1903	. . . . . . . . . 10 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
12		eleq1w 2817	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶))
13		dmeq 5904	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
14	13	eqeq1d 2735	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
15	12, 14	anbi12d 632	. . . . . . . . . . 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 283	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝜏 ↔ (𝑔 ∈ 𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))))
17	11, 16	sbiev 2309	. . . . . . . . 9 ⊢ ([𝑔 / 𝑓]𝜏 ↔ (𝑔 ∈ 𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
18	17	simplbi 499	. . . . . . . 8 ⊢ ([𝑔 / 𝑓]𝜏 → 𝑔 ∈ 𝐶)
19	18	3ad2ant3 1136	. . . . . . 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 2733	. . . . . . . 8 ⊢ (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
23	20, 21, 6, 22	bnj1326 34037	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐶 ∧ 𝑔 ∈ 𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
24	2, 5, 19, 23	syl3anc 1372	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
25	3	simprbi 498	. . . . . . . . . 10 ⊢ (𝜏 → dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
26	25	3ad2ant2 1135	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
27	17	simprbi 498	. . . . . . . . . 10 ⊢ ([𝑔 / 𝑓]𝜏 → dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
28	27	3ad2ant3 1136	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
29	26, 28	eqtr4d 2776	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑓 = dom 𝑔)
30		bnj1322 33833	. . . . . . . . 9 ⊢ (dom 𝑓 = dom 𝑔 → (dom 𝑓 ∩ dom 𝑔) = dom 𝑓)
31	30	reseq2d 5982	. . . . . . . 8 ⊢ (dom 𝑓 = dom 𝑔 → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑓 ↾ dom 𝑓))
32	29, 31	syl 17	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑓 ↾ dom 𝑓))
33		releq 5777	. . . . . . . . 9 ⊢ (𝑧 = 𝑓 → (Rel 𝑧 ↔ Rel 𝑓))
34	20, 21, 6	bnj66 33871	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐶 → Rel 𝑧)
35	33, 34	vtoclga 3566	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐶 → Rel 𝑓)
36		resdm 6027	. . . . . . . 8 ⊢ (Rel 𝑓 → (𝑓 ↾ dom 𝑓) = 𝑓)
37	5, 35, 36	3syl 18	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ dom 𝑓) = 𝑓)
38	32, 37	eqtrd 2773	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = 𝑓)
39		eqeq2 2745	. . . . . . . . . 10 ⊢ (dom 𝑓 = dom 𝑔 → ((dom 𝑓 ∩ dom 𝑔) = dom 𝑓 ↔ (dom 𝑓 ∩ dom 𝑔) = dom 𝑔))
40	30, 39	mpbid 231	. . . . . . . . 9 ⊢ (dom 𝑓 = dom 𝑔 → (dom 𝑓 ∩ dom 𝑔) = dom 𝑔)
41	40	reseq2d 5982	. . . . . . . 8 ⊢ (dom 𝑓 = dom 𝑔 → (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ dom 𝑔))
42	29, 41	syl 17	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ dom 𝑔))
43	20, 21, 6	bnj66 33871	. . . . . . . 8 ⊢ (𝑔 ∈ 𝐶 → Rel 𝑔)
44		resdm 6027	. . . . . . . 8 ⊢ (Rel 𝑔 → (𝑔 ↾ dom 𝑔) = 𝑔)
45	19, 43, 44	3syl 18	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ dom 𝑔) = 𝑔)
46	42, 45	eqtrd 2773	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)) = 𝑔)
47	24, 38, 46	3eqtr3d 2781	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔)
48	47	3expib 1123	. . . 4 ⊢ (𝑅 FrSe 𝐴 → ((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
49	48	alrimivv 1932	. . 3 ⊢ (𝑅 FrSe 𝐴 → ∀𝑓∀𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
50	49	adantr 482	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∀𝑓∀𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
51		nfv 1918	. . 3 ⊢ Ⅎ𝑔𝜏
52	51	eu2 2606	. 2 ⊢ (∃!𝑓𝜏 ↔ (∃𝑓𝜏 ∧ ∀𝑓∀𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔)))
53	1, 50, 52	sylanbrc 584	1 ⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃!𝑓𝜏)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088  ∀wal 1540   = wceq 1542  ∃wex 1782  [wsb 2068   ∈ wcel 2107  ∃!weu 2563  {cab 2710  ∀wral 3062  ∃wrex 3071   ∪ cun 3947   ∩ cin 3948   ⊆ wss 3949  {csn 4629  ⟨cop 4635  dom cdm 5677   ↾ cres 5679  Rel wrel 5682   Fn wfn 6539  ‘cfv 6544   predc-bnj14 33699   FrSe w-bnj15 33703   trClc-bnj18 33705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-reg 9587  ax-inf2 9636

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-bnj17 33698  df-bnj14 33700  df-bnj13 33702  df-bnj15 33704  df-bnj18 33706  df-bnj19 33708

This theorem is referenced by:  bnj1489  34067