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Theorem bnj1501 30878
Description: Technical lemma for bnj1500 30879. 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 480 . . . . . . . 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 30873 . . . . . . . . . 10 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
8 fndm 5953 . . . . . . . . . 10 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
97, 8syl 17 . . . . . . . . 9 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
101, 9bnj832 30571 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝐴)
112, 10eleqtrrd 2701 . . . . . . 7 (𝜑𝑥 ∈ dom 𝐹)
126dmeqi 5290 . . . . . . . 8 dom 𝐹 = dom 𝐶
135bnj1317 30635 . . . . . . . . 9 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
1413bnj1400 30649 . . . . . . . 8 dom 𝐶 = 𝑓𝐶 dom 𝑓
1512, 14eqtri 2643 . . . . . . 7 dom 𝐹 = 𝑓𝐶 dom 𝑓
1611, 15syl6eleq 2708 . . . . . 6 (𝜑𝑥 𝑓𝐶 dom 𝑓)
1716bnj1405 30650 . . . . 5 (𝜑 → ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
18 bnj1501.6 . . . . 5 (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))
1917, 18bnj1209 30610 . . . 4 (𝜑 → ∃𝑓𝜓)
205bnj1436 30653 . . . . . . . . . 10 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
2120bnj1299 30632 . . . . . . . . 9 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
22 fndm 5953 . . . . . . . . 9 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
2321, 22bnj31 30528 . . . . . . . 8 (𝑓𝐶 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
2418, 23bnj836 30573 . . . . . . 7 (𝜓 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
25 bnj1501.7 . . . . . . 7 (𝜒 ↔ (𝜓𝑑𝐵 ∧ dom 𝑓 = 𝑑))
263, 4, 5, 6, 1, 18bnj1518 30875 . . . . . . 7 (𝜓 → ∀𝑑𝜓)
2724, 25, 26bnj1521 30664 . . . . . 6 (𝜓 → ∃𝑑𝜒)
287bnj930 30583 . . . . . . . . . . . 12 (𝑅 FrSe 𝐴 → Fun 𝐹)
291, 28bnj832 30571 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
3018, 29bnj835 30572 . . . . . . . . . 10 (𝜓 → Fun 𝐹)
31 elssuni 4438 . . . . . . . . . . . 12 (𝑓𝐶𝑓 𝐶)
3231, 6syl6sseqr 3636 . . . . . . . . . . 11 (𝑓𝐶𝑓𝐹)
3318, 32bnj836 30573 . . . . . . . . . 10 (𝜓𝑓𝐹)
3418simp3bi 1076 . . . . . . . . . 10 (𝜓𝑥 ∈ dom 𝑓)
3530, 33, 34bnj1502 30661 . . . . . . . . 9 (𝜓 → (𝐹𝑥) = (𝑓𝑥))
363, 4, 5bnj1514 30874 . . . . . . . . . . 11 (𝑓𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3718, 36bnj836 30573 . . . . . . . . . 10 (𝜓 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3837, 34bnj1294 30631 . . . . . . . . 9 (𝜓 → (𝑓𝑥) = (𝐺𝑌))
3935, 38eqtrd 2655 . . . . . . . 8 (𝜓 → (𝐹𝑥) = (𝐺𝑌))
4025, 39bnj835 30572 . . . . . . 7 (𝜒 → (𝐹𝑥) = (𝐺𝑌))
4125, 30bnj835 30572 . . . . . . . . . . 11 (𝜒 → Fun 𝐹)
4225, 33bnj835 30572 . . . . . . . . . . 11 (𝜒𝑓𝐹)
433bnj1517 30663 . . . . . . . . . . . . . 14 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4425, 43bnj836 30573 . . . . . . . . . . . . 13 (𝜒 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4525, 34bnj835 30572 . . . . . . . . . . . . . 14 (𝜒𝑥 ∈ dom 𝑓)
4625simp3bi 1076 . . . . . . . . . . . . . 14 (𝜒 → dom 𝑓 = 𝑑)
4745, 46eleqtrd 2700 . . . . . . . . . . . . 13 (𝜒𝑥𝑑)
4844, 47bnj1294 30631 . . . . . . . . . . . 12 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4948, 46sseqtr4d 3626 . . . . . . . . . . 11 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑓)
5041, 42, 49bnj1503 30662 . . . . . . . . . 10 (𝜒 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
5150opeq2d 4382 . . . . . . . . 9 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
5251, 4syl6eqr 2673 . . . . . . . 8 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
5352fveq2d 6157 . . . . . . 7 (𝜒 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺𝑌))
5440, 53eqtr4d 2658 . . . . . 6 (𝜒 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5527, 54bnj593 30558 . . . . 5 (𝜓 → ∃𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
563, 4, 5, 6bnj1519 30876 . . . . 5 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5755, 56bnj1397 30648 . . . 4 (𝜓 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5819, 57bnj593 30558 . . 3 (𝜑 → ∃𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
593, 4, 5, 6bnj1520 30877 . . 3 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
6058, 59bnj1397 30648 . 2 (𝜑 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
611, 60bnj1459 30656 1 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  wss 3559  cop 4159   cuni 4407   ciun 4490  dom cdm 5079  cres 5081  Fun wfun 5846   Fn wfn 5847  cfv 5852   predc-bnj14 30496   FrSe w-bnj15 30500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-reg 8449  ax-inf2 8490
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-1o 7512  df-bnj17 30495  df-bnj14 30497  df-bnj13 30499  df-bnj15 30501  df-bnj18 30503  df-bnj19 30505
This theorem is referenced by:  bnj1500  30879
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