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Theorem bnj1501 31472
Description: Technical lemma for bnj1500 31473. 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 484 . . . . . . . 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 31467 . . . . . . . . . 10 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
8 fndm 6129 . . . . . . . . . 10 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
97, 8syl 17 . . . . . . . . 9 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
101, 9bnj832 31165 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝐴)
112, 10eleqtrrd 2853 . . . . . . 7 (𝜑𝑥 ∈ dom 𝐹)
126dmeqi 5462 . . . . . . . 8 dom 𝐹 = dom 𝐶
135bnj1317 31229 . . . . . . . . 9 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
1413bnj1400 31243 . . . . . . . 8 dom 𝐶 = 𝑓𝐶 dom 𝑓
1512, 14eqtri 2793 . . . . . . 7 dom 𝐹 = 𝑓𝐶 dom 𝑓
1611, 15syl6eleq 2860 . . . . . 6 (𝜑𝑥 𝑓𝐶 dom 𝑓)
1716bnj1405 31244 . . . . 5 (𝜑 → ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
18 bnj1501.6 . . . . 5 (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))
1917, 18bnj1209 31204 . . . 4 (𝜑 → ∃𝑓𝜓)
205bnj1436 31247 . . . . . . . . . 10 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
2120bnj1299 31226 . . . . . . . . 9 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
22 fndm 6129 . . . . . . . . 9 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
2321, 22bnj31 31124 . . . . . . . 8 (𝑓𝐶 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
2418, 23bnj836 31167 . . . . . . 7 (𝜓 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
25 bnj1501.7 . . . . . . 7 (𝜒 ↔ (𝜓𝑑𝐵 ∧ dom 𝑓 = 𝑑))
263, 4, 5, 6, 1, 18bnj1518 31469 . . . . . . 7 (𝜓 → ∀𝑑𝜓)
2724, 25, 26bnj1521 31258 . . . . . 6 (𝜓 → ∃𝑑𝜒)
287bnj930 31177 . . . . . . . . . . . 12 (𝑅 FrSe 𝐴 → Fun 𝐹)
291, 28bnj832 31165 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
3018, 29bnj835 31166 . . . . . . . . . 10 (𝜓 → Fun 𝐹)
31 elssuni 4604 . . . . . . . . . . . 12 (𝑓𝐶𝑓 𝐶)
3231, 6syl6sseqr 3801 . . . . . . . . . . 11 (𝑓𝐶𝑓𝐹)
3318, 32bnj836 31167 . . . . . . . . . 10 (𝜓𝑓𝐹)
3418simp3bi 1141 . . . . . . . . . 10 (𝜓𝑥 ∈ dom 𝑓)
3530, 33, 34bnj1502 31255 . . . . . . . . 9 (𝜓 → (𝐹𝑥) = (𝑓𝑥))
363, 4, 5bnj1514 31468 . . . . . . . . . . 11 (𝑓𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3718, 36bnj836 31167 . . . . . . . . . 10 (𝜓 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3837, 34bnj1294 31225 . . . . . . . . 9 (𝜓 → (𝑓𝑥) = (𝐺𝑌))
3935, 38eqtrd 2805 . . . . . . . 8 (𝜓 → (𝐹𝑥) = (𝐺𝑌))
4025, 39bnj835 31166 . . . . . . 7 (𝜒 → (𝐹𝑥) = (𝐺𝑌))
4125, 30bnj835 31166 . . . . . . . . . . 11 (𝜒 → Fun 𝐹)
4225, 33bnj835 31166 . . . . . . . . . . 11 (𝜒𝑓𝐹)
433bnj1517 31257 . . . . . . . . . . . . . 14 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4425, 43bnj836 31167 . . . . . . . . . . . . 13 (𝜒 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4525, 34bnj835 31166 . . . . . . . . . . . . . 14 (𝜒𝑥 ∈ dom 𝑓)
4625simp3bi 1141 . . . . . . . . . . . . . 14 (𝜒 → dom 𝑓 = 𝑑)
4745, 46eleqtrd 2852 . . . . . . . . . . . . 13 (𝜒𝑥𝑑)
4844, 47bnj1294 31225 . . . . . . . . . . . 12 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4948, 46sseqtr4d 3791 . . . . . . . . . . 11 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑓)
5041, 42, 49bnj1503 31256 . . . . . . . . . 10 (𝜒 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
5150opeq2d 4547 . . . . . . . . 9 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
5251, 4syl6eqr 2823 . . . . . . . 8 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
5352fveq2d 6337 . . . . . . 7 (𝜒 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺𝑌))
5440, 53eqtr4d 2808 . . . . . 6 (𝜒 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5527, 54bnj593 31152 . . . . 5 (𝜓 → ∃𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
563, 4, 5, 6bnj1519 31470 . . . . 5 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5755, 56bnj1397 31242 . . . 4 (𝜓 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5819, 57bnj593 31152 . . 3 (𝜑 → ∃𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
593, 4, 5, 6bnj1520 31471 . . 3 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
6058, 59bnj1397 31242 . 2 (𝜑 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
611, 60bnj1459 31250 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 382  w3a 1071   = wceq 1631  wcel 2145  {cab 2757  wral 3061  wrex 3062  wss 3723  cop 4323   cuni 4575   ciun 4655  dom cdm 5250  cres 5252  Fun wfun 6024   Fn wfn 6025  cfv 6030   predc-bnj14 31093   FrSe w-bnj15 31097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099  ax-reg 8656  ax-inf2 8705
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-om 7216  df-1o 7716  df-bnj17 31092  df-bnj14 31094  df-bnj13 31096  df-bnj15 31098  df-bnj18 31100  df-bnj19 31102
This theorem is referenced by:  bnj1500  31473
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