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Theorem bnj1501 31463
Description: Technical lemma for bnj1500 31464. 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 483 . . . . . . . 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 31458 . . . . . . . . . 10 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
8 fndm 6151 . . . . . . . . . 10 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
97, 8syl 17 . . . . . . . . 9 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
101, 9bnj832 31156 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝐴)
112, 10eleqtrrd 2842 . . . . . . 7 (𝜑𝑥 ∈ dom 𝐹)
126dmeqi 5480 . . . . . . . 8 dom 𝐹 = dom 𝐶
135bnj1317 31220 . . . . . . . . 9 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
1413bnj1400 31234 . . . . . . . 8 dom 𝐶 = 𝑓𝐶 dom 𝑓
1512, 14eqtri 2782 . . . . . . 7 dom 𝐹 = 𝑓𝐶 dom 𝑓
1611, 15syl6eleq 2849 . . . . . 6 (𝜑𝑥 𝑓𝐶 dom 𝑓)
1716bnj1405 31235 . . . . 5 (𝜑 → ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
18 bnj1501.6 . . . . 5 (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))
1917, 18bnj1209 31195 . . . 4 (𝜑 → ∃𝑓𝜓)
205bnj1436 31238 . . . . . . . . . 10 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
2120bnj1299 31217 . . . . . . . . 9 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
22 fndm 6151 . . . . . . . . 9 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
2321, 22bnj31 31115 . . . . . . . 8 (𝑓𝐶 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
2418, 23bnj836 31158 . . . . . . 7 (𝜓 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
25 bnj1501.7 . . . . . . 7 (𝜒 ↔ (𝜓𝑑𝐵 ∧ dom 𝑓 = 𝑑))
263, 4, 5, 6, 1, 18bnj1518 31460 . . . . . . 7 (𝜓 → ∀𝑑𝜓)
2724, 25, 26bnj1521 31249 . . . . . 6 (𝜓 → ∃𝑑𝜒)
287bnj930 31168 . . . . . . . . . . . 12 (𝑅 FrSe 𝐴 → Fun 𝐹)
291, 28bnj832 31156 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
3018, 29bnj835 31157 . . . . . . . . . 10 (𝜓 → Fun 𝐹)
31 elssuni 4619 . . . . . . . . . . . 12 (𝑓𝐶𝑓 𝐶)
3231, 6syl6sseqr 3793 . . . . . . . . . . 11 (𝑓𝐶𝑓𝐹)
3318, 32bnj836 31158 . . . . . . . . . 10 (𝜓𝑓𝐹)
3418simp3bi 1142 . . . . . . . . . 10 (𝜓𝑥 ∈ dom 𝑓)
3530, 33, 34bnj1502 31246 . . . . . . . . 9 (𝜓 → (𝐹𝑥) = (𝑓𝑥))
363, 4, 5bnj1514 31459 . . . . . . . . . . 11 (𝑓𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3718, 36bnj836 31158 . . . . . . . . . 10 (𝜓 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3837, 34bnj1294 31216 . . . . . . . . 9 (𝜓 → (𝑓𝑥) = (𝐺𝑌))
3935, 38eqtrd 2794 . . . . . . . 8 (𝜓 → (𝐹𝑥) = (𝐺𝑌))
4025, 39bnj835 31157 . . . . . . 7 (𝜒 → (𝐹𝑥) = (𝐺𝑌))
4125, 30bnj835 31157 . . . . . . . . . . 11 (𝜒 → Fun 𝐹)
4225, 33bnj835 31157 . . . . . . . . . . 11 (𝜒𝑓𝐹)
433bnj1517 31248 . . . . . . . . . . . . . 14 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4425, 43bnj836 31158 . . . . . . . . . . . . 13 (𝜒 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4525, 34bnj835 31157 . . . . . . . . . . . . . 14 (𝜒𝑥 ∈ dom 𝑓)
4625simp3bi 1142 . . . . . . . . . . . . . 14 (𝜒 → dom 𝑓 = 𝑑)
4745, 46eleqtrd 2841 . . . . . . . . . . . . 13 (𝜒𝑥𝑑)
4844, 47bnj1294 31216 . . . . . . . . . . . 12 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4948, 46sseqtr4d 3783 . . . . . . . . . . 11 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑓)
5041, 42, 49bnj1503 31247 . . . . . . . . . 10 (𝜒 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
5150opeq2d 4560 . . . . . . . . 9 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
5251, 4syl6eqr 2812 . . . . . . . 8 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
5352fveq2d 6357 . . . . . . 7 (𝜒 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺𝑌))
5440, 53eqtr4d 2797 . . . . . 6 (𝜒 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5527, 54bnj593 31143 . . . . 5 (𝜓 → ∃𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
563, 4, 5, 6bnj1519 31461 . . . . 5 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5755, 56bnj1397 31233 . . . 4 (𝜓 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5819, 57bnj593 31143 . . 3 (𝜑 → ∃𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
593, 4, 5, 6bnj1520 31462 . . 3 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
6058, 59bnj1397 31233 . 2 (𝜑 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
611, 60bnj1459 31241 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 383  w3a 1072   = wceq 1632  wcel 2139  {cab 2746  wral 3050  wrex 3051  wss 3715  cop 4327   cuni 4588   ciun 4672  dom cdm 5266  cres 5268  Fun wfun 6043   Fn wfn 6044  cfv 6049   predc-bnj14 31084   FrSe w-bnj15 31088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-reg 8664  ax-inf2 8713
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7232  df-1o 7730  df-bnj17 31083  df-bnj14 31085  df-bnj13 31087  df-bnj15 31089  df-bnj18 31091  df-bnj19 31093
This theorem is referenced by:  bnj1500  31464
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