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Theorem bnj1501 34831
Description: Technical lemma for bnj1500 34832. 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 34826 . . . . . . . . . 10 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
87fndmd 6660 . . . . . . . . 9 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
91, 8bnj832 34522 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝐴)
102, 9eleqtrrd 2828 . . . . . . 7 (𝜑𝑥 ∈ dom 𝐹)
116dmeqi 5907 . . . . . . . 8 dom 𝐹 = dom 𝐶
125bnj1317 34585 . . . . . . . . 9 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
1312bnj1400 34599 . . . . . . . 8 dom 𝐶 = 𝑓𝐶 dom 𝑓
1411, 13eqtri 2753 . . . . . . 7 dom 𝐹 = 𝑓𝐶 dom 𝑓
1510, 14eleqtrdi 2835 . . . . . 6 (𝜑𝑥 𝑓𝐶 dom 𝑓)
1615bnj1405 34600 . . . . 5 (𝜑 → ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
17 bnj1501.6 . . . . 5 (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))
1816, 17bnj1209 34560 . . . 4 (𝜑 → ∃𝑓𝜓)
195bnj1436 34603 . . . . . . . . . 10 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
2019bnj1299 34582 . . . . . . . . 9 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
21 fndm 6658 . . . . . . . . 9 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
2220, 21bnj31 34483 . . . . . . . 8 (𝑓𝐶 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
2317, 22bnj836 34524 . . . . . . 7 (𝜓 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
24 bnj1501.7 . . . . . . 7 (𝜒 ↔ (𝜓𝑑𝐵 ∧ dom 𝑓 = 𝑑))
253, 4, 5, 6, 1, 17bnj1518 34828 . . . . . . 7 (𝜓 → ∀𝑑𝜓)
2623, 24, 25bnj1521 34615 . . . . . 6 (𝜓 → ∃𝑑𝜒)
277fnfund 6656 . . . . . . . . . . . 12 (𝑅 FrSe 𝐴 → Fun 𝐹)
281, 27bnj832 34522 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
2917, 28bnj835 34523 . . . . . . . . . 10 (𝜓 → Fun 𝐹)
30 elssuni 4941 . . . . . . . . . . . 12 (𝑓𝐶𝑓 𝐶)
3130, 6sseqtrrdi 4028 . . . . . . . . . . 11 (𝑓𝐶𝑓𝐹)
3217, 31bnj836 34524 . . . . . . . . . 10 (𝜓𝑓𝐹)
3317simp3bi 1144 . . . . . . . . . 10 (𝜓𝑥 ∈ dom 𝑓)
3429, 32, 33bnj1502 34612 . . . . . . . . 9 (𝜓 → (𝐹𝑥) = (𝑓𝑥))
353, 4, 5bnj1514 34827 . . . . . . . . . . 11 (𝑓𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3617, 35bnj836 34524 . . . . . . . . . 10 (𝜓 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3736, 33bnj1294 34581 . . . . . . . . 9 (𝜓 → (𝑓𝑥) = (𝐺𝑌))
3834, 37eqtrd 2765 . . . . . . . 8 (𝜓 → (𝐹𝑥) = (𝐺𝑌))
3924, 38bnj835 34523 . . . . . . 7 (𝜒 → (𝐹𝑥) = (𝐺𝑌))
4024, 29bnj835 34523 . . . . . . . . . . 11 (𝜒 → Fun 𝐹)
4124, 32bnj835 34523 . . . . . . . . . . 11 (𝜒𝑓𝐹)
423bnj1517 34614 . . . . . . . . . . . . . 14 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4324, 42bnj836 34524 . . . . . . . . . . . . 13 (𝜒 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4424, 33bnj835 34523 . . . . . . . . . . . . . 14 (𝜒𝑥 ∈ dom 𝑓)
4524simp3bi 1144 . . . . . . . . . . . . . 14 (𝜒 → dom 𝑓 = 𝑑)
4644, 45eleqtrd 2827 . . . . . . . . . . . . 13 (𝜒𝑥𝑑)
4743, 46bnj1294 34581 . . . . . . . . . . . 12 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4847, 45sseqtrrd 4018 . . . . . . . . . . 11 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑓)
4940, 41, 48bnj1503 34613 . . . . . . . . . 10 (𝜒 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
5049opeq2d 4882 . . . . . . . . 9 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
5150, 4eqtr4di 2783 . . . . . . . 8 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
5251fveq2d 6900 . . . . . . 7 (𝜒 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺𝑌))
5339, 52eqtr4d 2768 . . . . . 6 (𝜒 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5426, 53bnj593 34509 . . . . 5 (𝜓 → ∃𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
553, 4, 5, 6bnj1519 34829 . . . . 5 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5654, 55bnj1397 34598 . . . 4 (𝜓 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5718, 56bnj593 34509 . . 3 (𝜑 → ∃𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
583, 4, 5, 6bnj1520 34830 . . 3 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5957, 58bnj1397 34598 . 2 (𝜑 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
601, 59bnj1459 34607 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 1084   = wceq 1533  wcel 2098  {cab 2702  wral 3050  wrex 3059  wss 3944  cop 4636   cuni 4909   ciun 4997  dom cdm 5678  cres 5680  Fun wfun 6543   Fn wfn 6544  cfv 6549   predc-bnj14 34452   FrSe w-bnj15 34456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-reg 9622  ax-inf2 9671
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-om 7872  df-1o 8487  df-bnj17 34451  df-bnj14 34453  df-bnj13 34455  df-bnj15 34457  df-bnj18 34459  df-bnj19 34461
This theorem is referenced by:  bnj1500  34832
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