Theorem bnj1384 32304

Description: Technical lemma for bnj60 32334. 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 32302	. . . . 5 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	bnj1371 32301	. . . 4 ⊢ (𝑓 ∈ 𝐻 → Fun 𝑓)
13	12	rgen 3148	. . 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 32303	. . . . . 6 ⊢ (𝑓 ∈ 𝐻 → 𝑓 ∈ 𝐶)
16		nfab1 2979	. . . . . . . . . 10 ⊢ Ⅎ𝑓{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
17	9, 16	nfcxfr 2975	. . . . . . . . 9 ⊢ Ⅎ𝑓𝐻
18	17	nfcri 2971	. . . . . . . 8 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐻
19		nfab1 2979	. . . . . . . . . 10 ⊢ Ⅎ𝑓{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
20	3, 19	nfcxfr 2975	. . . . . . . . 9 ⊢ Ⅎ𝑓𝐶
21	20	nfcri 2971	. . . . . . . 8 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐶
22	18, 21	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶)
23		eleq1w 2895	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐻 ↔ 𝑔 ∈ 𝐻))
24		eleq1w 2895	. . . . . . . 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 2242	. . . . . 6 ⊢ (𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶)
27		eqid 2821	. . . . . . 7 ⊢ (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
28	1, 2, 3, 27	bnj1326 32298	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐶 ∧ 𝑔 ∈ 𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
29	14, 15, 26, 28	syl3an 1156	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐻 ∧ 𝑔 ∈ 𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
30	29	3expib 1118	. . . 4 ⊢ (𝑅 FrSe 𝐴 → ((𝑓 ∈ 𝐻 ∧ 𝑔 ∈ 𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))))
31	30	ralrimivv 3190	. . 3 ⊢ (𝑅 FrSe 𝐴 → ∀𝑓 ∈ 𝐻 ∀𝑔 ∈ 𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
32		biid 263	. . . 4 ⊢ (∀𝑓 ∈ 𝐻 Fun 𝑓 ↔ ∀𝑓 ∈ 𝐻 Fun 𝑓)
33		biid 263	. . . 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 32093	. . . 4 ⊢ (𝑧 ∈ 𝐻 → ∀𝑓 𝑧 ∈ 𝐻)
35	32, 27, 33, 34	bnj1386 32105	. . 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 589	. 2 ⊢ (𝑅 FrSe 𝐴 → Fun ∪ 𝐻)
37	10	funeqi 6376	. 2 ⊢ (Fun 𝑃 ↔ Fun ∪ 𝐻)
38	36, 37	sylibr 236	1 ⊢ (𝑅 FrSe 𝐴 → Fun 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2799   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  {crab 3142  [wsbc 3772   ∪ cun 3934   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  {csn 4567  ⟨cop 4573  ∪ cuni 4838   class class class wbr 5066  dom cdm 5555   ↾ cres 5557  Fun wfun 6349   Fn wfn 6350  ‘cfv 6355   predc-bnj14 31958   FrSe w-bnj15 31962   trClc-bnj18 31964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-reg 9056  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-om 7581  df-1o 8102  df-bnj17 31957  df-bnj14 31959  df-bnj13 31961  df-bnj15 31963  df-bnj18 31965  df-bnj19 31967

This theorem is referenced by:  bnj1312  32330