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Theorem bnj1501 31515
Description: Technical lemma for bnj1500 31516. 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 490 . . . . . . . 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 31510 . . . . . . . . . 10 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
8 fndm 6168 . . . . . . . . . 10 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
97, 8syl 17 . . . . . . . . 9 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
101, 9bnj832 31208 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝐴)
112, 10eleqtrrd 2847 . . . . . . 7 (𝜑𝑥 ∈ dom 𝐹)
126dmeqi 5493 . . . . . . . 8 dom 𝐹 = dom 𝐶
135bnj1317 31272 . . . . . . . . 9 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
1413bnj1400 31286 . . . . . . . 8 dom 𝐶 = 𝑓𝐶 dom 𝑓
1512, 14eqtri 2787 . . . . . . 7 dom 𝐹 = 𝑓𝐶 dom 𝑓
1611, 15syl6eleq 2854 . . . . . 6 (𝜑𝑥 𝑓𝐶 dom 𝑓)
1716bnj1405 31287 . . . . 5 (𝜑 → ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
18 bnj1501.6 . . . . 5 (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))
1917, 18bnj1209 31247 . . . 4 (𝜑 → ∃𝑓𝜓)
205bnj1436 31290 . . . . . . . . . 10 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
2120bnj1299 31269 . . . . . . . . 9 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
22 fndm 6168 . . . . . . . . 9 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
2321, 22bnj31 31168 . . . . . . . 8 (𝑓𝐶 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
2418, 23bnj836 31210 . . . . . . 7 (𝜓 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
25 bnj1501.7 . . . . . . 7 (𝜒 ↔ (𝜓𝑑𝐵 ∧ dom 𝑓 = 𝑑))
263, 4, 5, 6, 1, 18bnj1518 31512 . . . . . . 7 (𝜓 → ∀𝑑𝜓)
2724, 25, 26bnj1521 31301 . . . . . 6 (𝜓 → ∃𝑑𝜒)
287bnj930 31220 . . . . . . . . . . . 12 (𝑅 FrSe 𝐴 → Fun 𝐹)
291, 28bnj832 31208 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
3018, 29bnj835 31209 . . . . . . . . . 10 (𝜓 → Fun 𝐹)
31 elssuni 4625 . . . . . . . . . . . 12 (𝑓𝐶𝑓 𝐶)
3231, 6syl6sseqr 3812 . . . . . . . . . . 11 (𝑓𝐶𝑓𝐹)
3318, 32bnj836 31210 . . . . . . . . . 10 (𝜓𝑓𝐹)
3418simp3bi 1177 . . . . . . . . . 10 (𝜓𝑥 ∈ dom 𝑓)
3530, 33, 34bnj1502 31298 . . . . . . . . 9 (𝜓 → (𝐹𝑥) = (𝑓𝑥))
363, 4, 5bnj1514 31511 . . . . . . . . . . 11 (𝑓𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3718, 36bnj836 31210 . . . . . . . . . 10 (𝜓 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3837, 34bnj1294 31268 . . . . . . . . 9 (𝜓 → (𝑓𝑥) = (𝐺𝑌))
3935, 38eqtrd 2799 . . . . . . . 8 (𝜓 → (𝐹𝑥) = (𝐺𝑌))
4025, 39bnj835 31209 . . . . . . 7 (𝜒 → (𝐹𝑥) = (𝐺𝑌))
4125, 30bnj835 31209 . . . . . . . . . . 11 (𝜒 → Fun 𝐹)
4225, 33bnj835 31209 . . . . . . . . . . 11 (𝜒𝑓𝐹)
433bnj1517 31300 . . . . . . . . . . . . . 14 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4425, 43bnj836 31210 . . . . . . . . . . . . 13 (𝜒 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4525, 34bnj835 31209 . . . . . . . . . . . . . 14 (𝜒𝑥 ∈ dom 𝑓)
4625simp3bi 1177 . . . . . . . . . . . . . 14 (𝜒 → dom 𝑓 = 𝑑)
4745, 46eleqtrd 2846 . . . . . . . . . . . . 13 (𝜒𝑥𝑑)
4844, 47bnj1294 31268 . . . . . . . . . . . 12 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4948, 46sseqtr4d 3802 . . . . . . . . . . 11 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑓)
5041, 42, 49bnj1503 31299 . . . . . . . . . 10 (𝜒 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
5150opeq2d 4566 . . . . . . . . 9 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
5251, 4syl6eqr 2817 . . . . . . . 8 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
5352fveq2d 6379 . . . . . . 7 (𝜒 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺𝑌))
5440, 53eqtr4d 2802 . . . . . 6 (𝜒 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5527, 54bnj593 31195 . . . . 5 (𝜓 → ∃𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
563, 4, 5, 6bnj1519 31513 . . . . 5 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5755, 56bnj1397 31285 . . . 4 (𝜓 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5819, 57bnj593 31195 . . 3 (𝜑 → ∃𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
593, 4, 5, 6bnj1520 31514 . . 3 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
6058, 59bnj1397 31285 . 2 (𝜑 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
611, 60bnj1459 31293 1 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  {cab 2751  wral 3055  wrex 3056  wss 3732  cop 4340   cuni 4594   ciun 4676  dom cdm 5277  cres 5279  Fun wfun 6062   Fn wfn 6063  cfv 6068   predc-bnj14 31137   FrSe w-bnj15 31141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-reg 8704  ax-inf2 8753
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-1o 7764  df-bnj17 31136  df-bnj14 31138  df-bnj13 31140  df-bnj15 31142  df-bnj18 31144  df-bnj19 31146
This theorem is referenced by:  bnj1500  31516
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