Theorem bnj1421 34990

Description: Technical lemma for bnj60 35010. 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
bnj1421.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1421.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1421.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1421.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1421.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1421.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1421.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1421.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1421.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1421.10	⊢ 𝑃 = ∪ 𝐻
bnj1421.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1421.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
bnj1421.13	⊢ (𝜒 → Fun 𝑃)
bnj1421.14	⊢ (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
bnj1421.15	⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))

Assertion

Ref	Expression
bnj1421	⊢ (𝜒 → Fun 𝑄)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

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

Proof of Theorem bnj1421

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1421.13	. . . 4 ⊢ (𝜒 → Fun 𝑃)
2		vex 3467	. . . . 5 ⊢ 𝑥 ∈ V
3		fvex 6898	. . . . 5 ⊢ (𝐺‘𝑍) ∈ V
4	2, 3	funsn 6598	. . . 4 ⊢ Fun {⟨𝑥, (𝐺‘𝑍)⟩}
5	1, 4	jctir 520	. . 3 ⊢ (𝜒 → (Fun 𝑃 ∧ Fun {⟨𝑥, (𝐺‘𝑍)⟩}))
6		bnj1421.15	. . . . 5 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
7	3	dmsnop 6216	. . . . . 6 ⊢ dom {⟨𝑥, (𝐺‘𝑍)⟩} = {𝑥}
8	7	a1i 11	. . . . 5 ⊢ (𝜒 → dom {⟨𝑥, (𝐺‘𝑍)⟩} = {𝑥})
9	6, 8	ineq12d 4201	. . . 4 ⊢ (𝜒 → (dom 𝑃 ∩ dom {⟨𝑥, (𝐺‘𝑍)⟩}) = ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}))
10		bnj1421.7	. . . . . . 7 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
11		bnj1421.6	. . . . . . . 8 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
12	11	simplbi 497	. . . . . . 7 ⊢ (𝜓 → 𝑅 FrSe 𝐴)
13	10, 12	bnj835 34707	. . . . . 6 ⊢ (𝜒 → 𝑅 FrSe 𝐴)
14		biid 261	. . . . . . . 8 ⊢ (𝑅 FrSe 𝐴 ↔ 𝑅 FrSe 𝐴)
15		biid 261	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
16		biid 261	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)))
17		biid 261	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))))
18		eqid 2734	. . . . . . . 8 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∪ ∪ 𝑧 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑧, 𝐴, 𝑅)) = ( pred(𝑥, 𝐴, 𝑅) ∪ ∪ 𝑧 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑧, 𝐴, 𝑅))
19	14, 15, 16, 17, 18	bnj1417 34989	. . . . . . 7 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
20		disjsn 4691	. . . . . . . 8 ⊢ (( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
21	20	ralbii 3081	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
22	19, 21	sylibr 234	. . . . . 6 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
23	13, 22	syl 17	. . . . 5 ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
24		bnj1421.5	. . . . . 6 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
25	24, 10	bnj1212 34747	. . . . 5 ⊢ (𝜒 → 𝑥 ∈ 𝐴)
26	23, 25	bnj1294 34765	. . . 4 ⊢ (𝜒 → ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
27	9, 26	eqtrd 2769	. . 3 ⊢ (𝜒 → (dom 𝑃 ∩ dom {⟨𝑥, (𝐺‘𝑍)⟩}) = ∅)
28		funun 6591	. . 3 ⊢ (((Fun 𝑃 ∧ Fun {⟨𝑥, (𝐺‘𝑍)⟩}) ∧ (dom 𝑃 ∩ dom {⟨𝑥, (𝐺‘𝑍)⟩}) = ∅) → Fun (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}))
29	5, 27, 28	syl2anc 584	. 2 ⊢ (𝜒 → Fun (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}))
30		bnj1421.12	. . 3 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
31	30	funeqi 6566	. 2 ⊢ (Fun 𝑄 ↔ Fun (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}))
32	29, 31	sylibr 234	1 ⊢ (𝜒 → Fun 𝑄)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {cab 2712   ≠ wne 2931  ∀wral 3050  ∃wrex 3059  {crab 3419  [wsbc 3770   ∪ cun 3929   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  {csn 4606  ⟨cop 4612  ∪ cuni 4887  ∪ ciun 4971   class class class wbr 5123  dom cdm 5665   ↾ cres 5667  Fun wfun 6534   Fn wfn 6535  ‘cfv 6540   predc-bnj14 34636   FrSe w-bnj15 34640   trClc-bnj18 34642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7736  ax-reg 9613  ax-inf2 9662

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6493  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7869  df-1o 8487  df-bnj17 34635  df-bnj14 34637  df-bnj13 34639  df-bnj15 34641  df-bnj18 34643  df-bnj19 34645

This theorem is referenced by:  bnj1312  35006