Theorem bnj1384 35015

Description: Technical lemma for bnj60 35045. 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 35013	. . . . 5 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	bnj1371 35012	. . . 4 ⊢ (𝑓 ∈ 𝐻 → Fun 𝑓)
13	12	rgen 3046	. . 3 ⊢ ∀𝑓 ∈ 𝐻 Fun 𝑓
14		id 22	. . . . . 6 ⊢ (𝑅 FrSe 𝐴 → 𝑅 FrSe 𝐴)
15	1, 2, 3, 4, 5, 6, 7, 8, 9	bnj1374 35014	. . . . . 6 ⊢ (𝑓 ∈ 𝐻 → 𝑓 ∈ 𝐶)
16		nfab1 2893	. . . . . . . . . 10 ⊢ Ⅎ𝑓{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
17	9, 16	nfcxfr 2889	. . . . . . . . 9 ⊢ Ⅎ𝑓𝐻
18	17	nfcri 2883	. . . . . . . 8 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐻
19		nfab1 2893	. . . . . . . . . 10 ⊢ Ⅎ𝑓{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
20	3, 19	nfcxfr 2889	. . . . . . . . 9 ⊢ Ⅎ𝑓𝐶
21	20	nfcri 2883	. . . . . . . 8 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐶
22	18, 21	nfim 1896	. . . . . . 7 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶)
23		eleq1w 2811	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐻 ↔ 𝑔 ∈ 𝐻))
24		eleq1w 2811	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶))
25	23, 24	imbi12d 344	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((𝑓 ∈ 𝐻 → 𝑓 ∈ 𝐶) ↔ (𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶)))
26	22, 25, 15	chvarfv 2241	. . . . . 6 ⊢ (𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶)
27		eqid 2729	. . . . . . 7 ⊢ (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
28	1, 2, 3, 27	bnj1326 35009	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐶 ∧ 𝑔 ∈ 𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
29	14, 15, 26, 28	syl3an 1160	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐻 ∧ 𝑔 ∈ 𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
30	29	3expib 1122	. . . 4 ⊢ (𝑅 FrSe 𝐴 → ((𝑓 ∈ 𝐻 ∧ 𝑔 ∈ 𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))))
31	30	ralrimivv 3176	. . 3 ⊢ (𝑅 FrSe 𝐴 → ∀𝑓 ∈ 𝐻 ∀𝑔 ∈ 𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
32		biid 261	. . . 4 ⊢ (∀𝑓 ∈ 𝐻 Fun 𝑓 ↔ ∀𝑓 ∈ 𝐻 Fun 𝑓)
33		biid 261	. . . 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 34804	. . . 4 ⊢ (𝑧 ∈ 𝐻 → ∀𝑓 𝑧 ∈ 𝐻)
35	32, 27, 33, 34	bnj1386 34816	. . 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 587	. 2 ⊢ (𝑅 FrSe 𝐴 → Fun ∪ 𝐻)
37	10	funeqi 6521	. 2 ⊢ (Fun 𝑃 ↔ Fun ∪ 𝐻)
38	36, 37	sylibr 234	1 ⊢ (𝑅 FrSe 𝐴 → Fun 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3402  [wsbc 3750   ∪ cun 3909   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  {csn 4585  ⟨cop 4591  ∪ cuni 4867   class class class wbr 5102  dom cdm 5631   ↾ cres 5633  Fun wfun 6493   Fn wfn 6494  ‘cfv 6499   predc-bnj14 34671   FrSe w-bnj15 34675   trClc-bnj18 34677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-reg 9521  ax-inf2 9570

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-1o 8411  df-bnj17 34670  df-bnj14 34672  df-bnj13 34674  df-bnj15 34676  df-bnj18 34678  df-bnj19 34680

This theorem is referenced by:  bnj1312  35041