Theorem bnj1384 35429

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
bnj1384.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1384.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1384.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1384.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1384.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1384.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1384.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1384.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1384.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1384.10	⊢ 𝑃 = ∪ 𝐻

Assertion

Ref	Expression
bnj1384	⊢ (𝑅 FrSe 𝐴 → Fun 𝑃)

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

Allowed substitution hints:   𝜓(𝑥, 𝑦, 𝑓, 𝑑)   𝜒(𝑥, 𝑦, 𝑓, 𝑑)   𝜏(𝑥, 𝑦, 𝑓, 𝑑)   𝐴(𝑦)   𝐵(𝑥, 𝑦, 𝑑)   𝐶(𝑥, 𝑓, 𝑑)   𝐷(𝑥, 𝑦, 𝑓, 𝑑)   𝑃(𝑥, 𝑦, 𝑓, 𝑑)   𝑅(𝑦)   𝐺(𝑥, 𝑦)   𝐻(𝑥, 𝑦, 𝑓, 𝑑)   𝑌(𝑥, 𝑦, 𝑓, 𝑑)   𝜏′(𝑥, 𝑦, 𝑓, 𝑑)

Proof of Theorem bnj1384

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

Step	Hyp	Ref	Expression
1		bnj1384.1	. . . . 5 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2		bnj1384.2	. . . . 5 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
3		bnj1384.3	. . . . 5 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
4		bnj1384.4	. . . . 5 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
5		bnj1384.5	. . . . 5 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
6		bnj1384.6	. . . . 5 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
7		bnj1384.7	. . . . 5 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
8		bnj1384.8	. . . . 5 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
9		bnj1384.9	. . . . 5 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
10		bnj1384.10	. . . . 5 ⊢ 𝑃 = ∪ 𝐻
11	1, 2, 3, 4, 8	bnj1373 35427	. . . . 5 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	bnj1371 35426	. . . 4 ⊢ (𝑓 ∈ 𝐻 → Fun 𝑓)
13	12	rgen 3080	. . 3 ⊢ ∀𝑓 ∈ 𝐻 Fun 𝑓
14		id 23	. . . . . 6 ⊢ (𝑅 FrSe 𝐴 → 𝑅 FrSe 𝐴)
15	1, 2, 3, 4, 5, 6, 7, 8, 9	bnj1374 35428	. . . . . 6 ⊢ (𝑓 ∈ 𝐻 → 𝑓 ∈ 𝐶)
16		nfab1 2926	. . . . . . . . . 10 ⊢ Ⅎ𝑓{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
17	9, 16	nfcxfr 2922	. . . . . . . . 9 ⊢ Ⅎ𝑓𝐻
18	17	nfcri 2916	. . . . . . . 8 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐻
19		nfab1 2926	. . . . . . . . . 10 ⊢ Ⅎ𝑓{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
20	3, 19	nfcxfr 2922	. . . . . . . . 9 ⊢ Ⅎ𝑓𝐶
21	20	nfcri 2916	. . . . . . . 8 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐶
22	18, 21	nfim 1925	. . . . . . 7 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶)
23		eleq1w 2845	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐻 ↔ 𝑔 ∈ 𝐻))
24		eleq1w 2845	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶))
25	23, 24	imbi12d 347	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((𝑓 ∈ 𝐻 → 𝑓 ∈ 𝐶) ↔ (𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶)))
26	22, 25, 15	chvarfv 2275	. . . . . 6 ⊢ (𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶)
27		eqid 2762	. . . . . . 7 ⊢ (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
28	1, 2, 3, 27	bnj1326 35423	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐶 ∧ 𝑔 ∈ 𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
29	14, 15, 26, 28	syl3an 1177	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐻 ∧ 𝑔 ∈ 𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
30	29	3expib 1139	. . . 4 ⊢ (𝑅 FrSe 𝐴 → ((𝑓 ∈ 𝐻 ∧ 𝑔 ∈ 𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))))
31	30	ralrimivv 3205	. . 3 ⊢ (𝑅 FrSe 𝐴 → ∀𝑓 ∈ 𝐻 ∀𝑔 ∈ 𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
32		biid 264	. . . 4 ⊢ (∀𝑓 ∈ 𝐻 Fun 𝑓 ↔ ∀𝑓 ∈ 𝐻 Fun 𝑓)
33		biid 264	. . . 4 ⊢ ((∀𝑓 ∈ 𝐻 Fun 𝑓 ∧ ∀𝑓 ∈ 𝐻 ∀𝑔 ∈ 𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))) ↔ (∀𝑓 ∈ 𝐻 Fun 𝑓 ∧ ∀𝑓 ∈ 𝐻 ∀𝑔 ∈ 𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))))
34	9	bnj1317 35218	. . . 4 ⊢ (𝑧 ∈ 𝐻 → ∀𝑓 𝑧 ∈ 𝐻)
35	32, 27, 33, 34	bnj1386 35230	. . 3 ⊢ ((∀𝑓 ∈ 𝐻 Fun 𝑓 ∧ ∀𝑓 ∈ 𝐻 ∀𝑔 ∈ 𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))) → Fun ∪ 𝐻)
36	13, 31, 35	sylancr 598	. 2 ⊢ (𝑅 FrSe 𝐴 → Fun ∪ 𝐻)
37	10	funeqi 6557	. 2 ⊢ (Fun 𝑃 ↔ Fun ∪ 𝐻)
38	36, 37	sylibr 237	1 ⊢ (𝑅 FrSe 𝐴 → Fun 𝑃)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569  ∃wex 1808   ∈ wcel 2142  {cab 2740   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3415  [wsbc 3743   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {csn 4588  ⟨cop 4594  ∪ cuni 4871   class class class wbr 5108  dom cdm 5660   ↾ cres 5662  Fun wfun 6530   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:  bnj1312  35455