Theorem bnj1421 35739

Description: Technical lemma for bnj60 35759. 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 3458	. . . . 5 ⊢ 𝑥 ∈ V
3		fvex 6918	. . . . 5 ⊢ (𝐺‘𝑍) ∈ V
4	2, 3	funsn 6612	. . . 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 6229	. . . . . 6 ⊢ dom {⟨𝑥, (𝐺‘𝑍)⟩} = {𝑥}
8	7	a1i 11	. . . . 5 ⊢ (𝜒 → dom {⟨𝑥, (𝐺‘𝑍)⟩} = {𝑥})
9	6, 8	ineq12d 4121	. . . 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 35457	. . . . . 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 2763	. . . . . . . 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 35738	. . . . . . 7 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
20		disjsn 4672	. . . . . . . 8 ⊢ (( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
21	20	ralbii 3112	. . . . . . 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 35496	. . . . 5 ⊢ (𝜒 → 𝑥 ∈ 𝐴)
26	23, 25	bnj1294 35514	. . . 4 ⊢ (𝜒 → ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
27	9, 26	eqtrd 2798	. . 3 ⊢ (𝜒 → (dom 𝑃 ∩ dom {⟨𝑥, (𝐺‘𝑍)⟩}) = ∅)
28		funun 6603	. . 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 6577	. 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 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416  [wsbc 3742   ⊆ wss 3900   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588  ⟨cop 4838  ∪ cuni 4870  ∪ ciun 4954   class class class wbr 5106  dom cdm 5655   ↾ cres 5657  Fun wfun 6549   Fn wfn 6550  ‘cfv 6555   predc-bnj14 35386   FrSe w-bnj15 35390   trClc-bnj18 35392

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-reg 9610  ax-inf2 9666

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-om 7899  df-1o 8501  df-bnj17 35385  df-bnj14 35387  df-bnj13 35389  df-bnj15 35391  df-bnj18 35393  df-bnj19 35395

This theorem is used by:  bnj1312  35755