Theorem bnj1421 35443

Description: Technical lemma for bnj60 35463. 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 3462	. . . . 5 ⊢ 𝑥 ∈ V
3		fvex 6898	. . . . 5 ⊢ (𝐺‘𝑍) ∈ V
4	2, 3	funsn 6593	. . . 4 ⊢ Fun {⟨𝑥, (𝐺‘𝑍)⟩}
5	1, 4	jctir 530	. . 3 ⊢ (𝜒 → (Fun 𝑃 ∧ Fun {⟨𝑥, (𝐺‘𝑍)⟩}))
6		bnj1421.15	. . . . 5 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
7	3	dmsnop 6220	. . . . . 6 ⊢ dom {⟨𝑥, (𝐺‘𝑍)⟩} = {𝑥}
8	7	a1i 11	. . . . 5 ⊢ (𝜒 → dom {⟨𝑥, (𝐺‘𝑍)⟩} = {𝑥})
9	6, 8	ineq12d 4177	. . . 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 502	. . . . . . 7 ⊢ (𝜓 → 𝑅 FrSe 𝐴)
13	10, 12	bnj835 35161	. . . . . 6 ⊢ (𝜒 → 𝑅 FrSe 𝐴)
14		biid 264	. . . . . . . 8 ⊢ (𝑅 FrSe 𝐴 ↔ 𝑅 FrSe 𝐴)
15		biid 264	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
16		biid 264	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)))
17		biid 264	. . . . . . . 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 2766	. . . . . . . 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 35442	. . . . . . 7 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
20		disjsn 4680	. . . . . . . 8 ⊢ (( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
21	20	ralbii 3114	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
22	19, 21	sylibr 237	. . . . . 6 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
23	13, 22	syl 18	. . . . 5 ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
24		bnj1421.5	. . . . . 6 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
25	24, 10	bnj1212 35200	. . . . 5 ⊢ (𝜒 → 𝑥 ∈ 𝐴)
26	23, 25	bnj1294 35218	. . . 4 ⊢ (𝜒 → ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
27	9, 26	eqtrd 2801	. . 3 ⊢ (𝜒 → (dom 𝑃 ∩ dom {⟨𝑥, (𝐺‘𝑍)⟩}) = ∅)
28		funun 6586	. . 3 ⊢ (((Fun 𝑃 ∧ Fun {⟨𝑥, (𝐺‘𝑍)⟩}) ∧ (dom 𝑃 ∩ dom {⟨𝑥, (𝐺‘𝑍)⟩}) = ∅) → Fun (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}))
29	5, 27, 28	syl2anc 596	. 2 ⊢ (𝜒 → Fun (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}))
30		bnj1421.12	. . 3 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
31	30	funeqi 6561	. 2 ⊢ (Fun 𝑄 ↔ Fun (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}))
32	29, 31	sylibr 237	1 ⊢ (𝜒 → Fun 𝑄)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2744   ≠ wne 2961  ∀wral 3082  ∃wrex 3092  {crab 3419  [wsbc 3747   ∪ cun 3906   ∩ cin 3907   ⊆ wss 3908  ∅c0 4289  {csn 4592  ⟨cop 4598  ∪ cuni 4875  ∪ ciun 4959   class class class wbr 5112  dom cdm 5664   ↾ cres 5666  Fun wfun 6534   Fn wfn 6535  ‘cfv 6540   predc-bnj14 35090   FrSe w-bnj15 35094   trClc-bnj18 35096

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5241  ax-sep 5260  ax-nul 5272  ax-pow 5339  ax-pr 5407  ax-un 7738  ax-reg 9556  ax-inf2 9612

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4876  df-iun 4961  df-br 5113  df-opab 5177  df-mpt 5196  df-tr 5222  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7865  df-1o 8455  df-bnj17 35089  df-bnj14 35091  df-bnj13 35093  df-bnj15 35095  df-bnj18 35097  df-bnj19 35099

This theorem is used by:  bnj1312  35459