Theorem bnj1421 35200

Description: Technical lemma for bnj60 35220. 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 3445	. . . . 5 ⊢ 𝑥 ∈ V
3		fvex 6848	. . . . 5 ⊢ (𝐺‘𝑍) ∈ V
4	2, 3	funsn 6546	. . . 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 6175	. . . . . 6 ⊢ dom {⟨𝑥, (𝐺‘𝑍)⟩} = {𝑥}
8	7	a1i 11	. . . . 5 ⊢ (𝜒 → dom {⟨𝑥, (𝐺‘𝑍)⟩} = {𝑥})
9	6, 8	ineq12d 4174	. . . 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 34917	. . . . . 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 2737	. . . . . . . 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 35199	. . . . . . 7 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
20		disjsn 4669	. . . . . . . 8 ⊢ (( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
21	20	ralbii 3083	. . . . . . 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 34957	. . . . 5 ⊢ (𝜒 → 𝑥 ∈ 𝐴)
26	23, 25	bnj1294 34975	. . . 4 ⊢ (𝜒 → ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
27	9, 26	eqtrd 2772	. . 3 ⊢ (𝜒 → (dom 𝑃 ∩ dom {⟨𝑥, (𝐺‘𝑍)⟩}) = ∅)
28		funun 6539	. . 3 ⊢ (((Fun 𝑃 ∧ Fun {⟨𝑥, (𝐺‘𝑍)⟩}) ∧ (dom 𝑃 ∩ dom {⟨𝑥, (𝐺‘𝑍)⟩}) = ∅) → Fun (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}))
29	5, 27, 28	syl2anc 585	. 2 ⊢ (𝜒 → Fun (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}))
30		bnj1421.12	. . 3 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
31	30	funeqi 6514	. 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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  {crab 3400  [wsbc 3741   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4286  {csn 4581  ⟨cop 4587  ∪ cuni 4864  ∪ ciun 4947   class class class wbr 5099  dom cdm 5625   ↾ cres 5627  Fun wfun 6487   Fn wfn 6488  ‘cfv 6493   predc-bnj14 34846   FrSe w-bnj15 34850   trClc-bnj18 34852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-reg 9501  ax-inf2 9554

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-om 7811  df-1o 8399  df-bnj17 34845  df-bnj14 34847  df-bnj13 34849  df-bnj15 34851  df-bnj18 34853  df-bnj19 34855

This theorem is referenced by:  bnj1312  35216