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Theorem bnj1501 34376
Description: Technical lemma for bnj1500 34377. 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
bnj1501.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1501.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1501.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1501.4 𝐹 = 𝐶
bnj1501.5 (𝜑 ↔ (𝑅 FrSe 𝐴𝑥𝐴))
bnj1501.6 (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))
bnj1501.7 (𝜒 ↔ (𝜓𝑑𝐵 ∧ dom 𝑓 = 𝑑))
Assertion
Ref Expression
bnj1501 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝐺,𝑑,𝑓,𝑥   𝑅,𝑑,𝑓,𝑥   𝑌,𝑑   𝜑,𝑑,𝑓
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑓,𝑑)   𝜒(𝑥,𝑓,𝑑)   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝐹(𝑥,𝑓,𝑑)   𝑌(𝑥,𝑓)

Proof of Theorem bnj1501
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 bnj1501.5 . 2 (𝜑 ↔ (𝑅 FrSe 𝐴𝑥𝐴))
21simprbi 495 . . . . . . . 8 (𝜑𝑥𝐴)
3 bnj1501.1 . . . . . . . . . . 11 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
4 bnj1501.2 . . . . . . . . . . 11 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
5 bnj1501.3 . . . . . . . . . . 11 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
6 bnj1501.4 . . . . . . . . . . 11 𝐹 = 𝐶
73, 4, 5, 6bnj60 34371 . . . . . . . . . 10 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
87fndmd 6653 . . . . . . . . 9 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
91, 8bnj832 34067 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝐴)
102, 9eleqtrrd 2834 . . . . . . 7 (𝜑𝑥 ∈ dom 𝐹)
116dmeqi 5903 . . . . . . . 8 dom 𝐹 = dom 𝐶
125bnj1317 34130 . . . . . . . . 9 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
1312bnj1400 34144 . . . . . . . 8 dom 𝐶 = 𝑓𝐶 dom 𝑓
1411, 13eqtri 2758 . . . . . . 7 dom 𝐹 = 𝑓𝐶 dom 𝑓
1510, 14eleqtrdi 2841 . . . . . 6 (𝜑𝑥 𝑓𝐶 dom 𝑓)
1615bnj1405 34145 . . . . 5 (𝜑 → ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
17 bnj1501.6 . . . . 5 (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))
1816, 17bnj1209 34105 . . . 4 (𝜑 → ∃𝑓𝜓)
195bnj1436 34148 . . . . . . . . . 10 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
2019bnj1299 34127 . . . . . . . . 9 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
21 fndm 6651 . . . . . . . . 9 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
2220, 21bnj31 34028 . . . . . . . 8 (𝑓𝐶 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
2317, 22bnj836 34069 . . . . . . 7 (𝜓 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
24 bnj1501.7 . . . . . . 7 (𝜒 ↔ (𝜓𝑑𝐵 ∧ dom 𝑓 = 𝑑))
253, 4, 5, 6, 1, 17bnj1518 34373 . . . . . . 7 (𝜓 → ∀𝑑𝜓)
2623, 24, 25bnj1521 34160 . . . . . 6 (𝜓 → ∃𝑑𝜒)
277fnfund 6649 . . . . . . . . . . . 12 (𝑅 FrSe 𝐴 → Fun 𝐹)
281, 27bnj832 34067 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
2917, 28bnj835 34068 . . . . . . . . . 10 (𝜓 → Fun 𝐹)
30 elssuni 4940 . . . . . . . . . . . 12 (𝑓𝐶𝑓 𝐶)
3130, 6sseqtrrdi 4032 . . . . . . . . . . 11 (𝑓𝐶𝑓𝐹)
3217, 31bnj836 34069 . . . . . . . . . 10 (𝜓𝑓𝐹)
3317simp3bi 1145 . . . . . . . . . 10 (𝜓𝑥 ∈ dom 𝑓)
3429, 32, 33bnj1502 34157 . . . . . . . . 9 (𝜓 → (𝐹𝑥) = (𝑓𝑥))
353, 4, 5bnj1514 34372 . . . . . . . . . . 11 (𝑓𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3617, 35bnj836 34069 . . . . . . . . . 10 (𝜓 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3736, 33bnj1294 34126 . . . . . . . . 9 (𝜓 → (𝑓𝑥) = (𝐺𝑌))
3834, 37eqtrd 2770 . . . . . . . 8 (𝜓 → (𝐹𝑥) = (𝐺𝑌))
3924, 38bnj835 34068 . . . . . . 7 (𝜒 → (𝐹𝑥) = (𝐺𝑌))
4024, 29bnj835 34068 . . . . . . . . . . 11 (𝜒 → Fun 𝐹)
4124, 32bnj835 34068 . . . . . . . . . . 11 (𝜒𝑓𝐹)
423bnj1517 34159 . . . . . . . . . . . . . 14 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4324, 42bnj836 34069 . . . . . . . . . . . . 13 (𝜒 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4424, 33bnj835 34068 . . . . . . . . . . . . . 14 (𝜒𝑥 ∈ dom 𝑓)
4524simp3bi 1145 . . . . . . . . . . . . . 14 (𝜒 → dom 𝑓 = 𝑑)
4644, 45eleqtrd 2833 . . . . . . . . . . . . 13 (𝜒𝑥𝑑)
4743, 46bnj1294 34126 . . . . . . . . . . . 12 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4847, 45sseqtrrd 4022 . . . . . . . . . . 11 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑓)
4940, 41, 48bnj1503 34158 . . . . . . . . . 10 (𝜒 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
5049opeq2d 4879 . . . . . . . . 9 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
5150, 4eqtr4di 2788 . . . . . . . 8 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
5251fveq2d 6894 . . . . . . 7 (𝜒 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺𝑌))
5339, 52eqtr4d 2773 . . . . . 6 (𝜒 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5426, 53bnj593 34054 . . . . 5 (𝜓 → ∃𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
553, 4, 5, 6bnj1519 34374 . . . . 5 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5654, 55bnj1397 34143 . . . 4 (𝜓 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5718, 56bnj593 34054 . . 3 (𝜑 → ∃𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
583, 4, 5, 6bnj1520 34375 . . 3 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5957, 58bnj1397 34143 . 2 (𝜑 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
601, 59bnj1459 34152 1 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1085   = wceq 1539  wcel 2104  {cab 2707  wral 3059  wrex 3068  wss 3947  cop 4633   cuni 4907   ciun 4996  dom cdm 5675  cres 5677  Fun wfun 6536   Fn wfn 6537  cfv 6542   predc-bnj14 33997   FrSe w-bnj15 34001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-reg 9589  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-1o 8468  df-bnj17 33996  df-bnj14 33998  df-bnj13 34000  df-bnj15 34002  df-bnj18 34004  df-bnj19 34006
This theorem is referenced by:  bnj1500  34377
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