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Theorem bnj1421 32595
Description: Technical lemma for bnj60 32615. 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 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1421.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1421.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1421.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1421.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1421.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1421.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 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, (𝐺𝑍)⟩})
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.
StepHypRef Expression
1 bnj1421.13 . . . 4 (𝜒 → Fun 𝑃)
2 vex 3402 . . . . 5 𝑥 ∈ V
3 fvex 6689 . . . . 5 (𝐺𝑍) ∈ V
42, 3funsn 6392 . . . 4 Fun {⟨𝑥, (𝐺𝑍)⟩}
51, 4jctir 524 . . 3 (𝜒 → (Fun 𝑃 ∧ Fun {⟨𝑥, (𝐺𝑍)⟩}))
6 bnj1421.15 . . . . 5 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
73dmsnop 6048 . . . . . 6 dom {⟨𝑥, (𝐺𝑍)⟩} = {𝑥}
87a1i 11 . . . . 5 (𝜒 → dom {⟨𝑥, (𝐺𝑍)⟩} = {𝑥})
96, 8ineq12d 4104 . . . 4 (𝜒 → (dom 𝑃 ∩ dom {⟨𝑥, (𝐺𝑍)⟩}) = ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}))
10 bnj1421.7 . . . . . . 7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
11 bnj1421.6 . . . . . . . 8 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
1211simplbi 501 . . . . . . 7 (𝜓𝑅 FrSe 𝐴)
1310, 12bnj835 32311 . . . . . 6 (𝜒𝑅 FrSe 𝐴)
14 biid 264 . . . . . . . 8 (𝑅 FrSe 𝐴𝑅 FrSe 𝐴)
15 biid 264 . . . . . . . 8 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
16 biid 264 . . . . . . . 8 (∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)))
17 biid 264 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴 ∧ ∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑅 FrSe 𝐴𝑥𝐴 ∧ ∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))))
18 eqid 2738 . . . . . . . 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))
1914, 15, 16, 17, 18bnj1417 32594 . . . . . . 7 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
20 disjsn 4602 . . . . . . . 8 (( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
2120ralbii 3080 . . . . . . 7 (∀𝑥𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
2219, 21sylibr 237 . . . . . 6 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
2313, 22syl 17 . . . . 5 (𝜒 → ∀𝑥𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
24 bnj1421.5 . . . . . 6 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
2524, 10bnj1212 32352 . . . . 5 (𝜒𝑥𝐴)
2623, 25bnj1294 32370 . . . 4 (𝜒 → ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
279, 26eqtrd 2773 . . 3 (𝜒 → (dom 𝑃 ∩ dom {⟨𝑥, (𝐺𝑍)⟩}) = ∅)
28 funun 6385 . . 3 (((Fun 𝑃 ∧ Fun {⟨𝑥, (𝐺𝑍)⟩}) ∧ (dom 𝑃 ∩ dom {⟨𝑥, (𝐺𝑍)⟩}) = ∅) → Fun (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
295, 27, 28syl2anc 587 . 2 (𝜒 → Fun (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
30 bnj1421.12 . . 3 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
3130funeqi 6360 . 2 (Fun 𝑄 ↔ Fun (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
3229, 31sylibr 237 1 (𝜒 → Fun 𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wex 1786  wcel 2114  {cab 2716  wne 2934  wral 3053  wrex 3054  {crab 3057  [wsbc 3680  cun 3841  cin 3842  wss 3843  c0 4211  {csn 4516  cop 4522   cuni 4796   ciun 4881   class class class wbr 5030  dom cdm 5525  cres 5527  Fun wfun 6333   Fn wfn 6334  cfv 6339   predc-bnj14 32239   FrSe w-bnj15 32243   trClc-bnj18 32245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481  ax-reg 9131  ax-inf2 9179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7602  df-1o 8133  df-bnj17 32238  df-bnj14 32240  df-bnj13 32242  df-bnj15 32244  df-bnj18 32246  df-bnj19 32248
This theorem is referenced by:  bnj1312  32611
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