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Theorem bnj1286 34343
Description: Technical lemma for bnj60 34386. 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
bnj1286.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1286.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1286.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1286.4 𝐷 = (dom 𝑔 ∩ dom )
bnj1286.5 𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
bnj1286.6 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
bnj1286.7 (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))
Assertion
Ref Expression
bnj1286 (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷)
Distinct variable groups:   𝐴,𝑑,𝑓   𝐵,𝑓,𝑔   𝐵,,𝑓   𝑥,𝐷   𝑓,𝐺,𝑔   ,𝐺   𝑅,𝑑,𝑓   𝑔,𝑌   ,𝑌   𝑔,𝑑,𝑥,𝑓   ,𝑑,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝜓(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐴(𝑥,𝑦,𝑔,)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐷(𝑦,𝑓,𝑔,,𝑑)   𝑅(𝑥,𝑦,𝑔,)   𝐸(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐺(𝑥,𝑦,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1286
StepHypRef Expression
1 bnj1286.7 . . . . 5 (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))
2 bnj1286.1 . . . . . . . . 9 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
3 bnj1286.2 . . . . . . . . 9 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
4 bnj1286.3 . . . . . . . . 9 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
5 bnj1286.4 . . . . . . . . 9 𝐷 = (dom 𝑔 ∩ dom )
6 bnj1286.5 . . . . . . . . 9 𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
7 bnj1286.6 . . . . . . . . 9 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
82, 3, 4, 5, 6, 7, 1bnj1256 34339 . . . . . . . 8 (𝜑 → ∃𝑑𝐵 𝑔 Fn 𝑑)
98bnj1196 34118 . . . . . . 7 (𝜑 → ∃𝑑(𝑑𝐵𝑔 Fn 𝑑))
102bnj1517 34174 . . . . . . . . 9 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
1110adantr 480 . . . . . . . 8 ((𝑑𝐵𝑔 Fn 𝑑) → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
12 fndm 6652 . . . . . . . . . 10 (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
13 sseq2 4008 . . . . . . . . . . 11 (dom 𝑔 = 𝑑 → ( pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔 ↔ pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
1413raleqbi1dv 3332 . . . . . . . . . 10 (dom 𝑔 = 𝑑 → (∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔 ↔ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
1512, 14syl 17 . . . . . . . . 9 (𝑔 Fn 𝑑 → (∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔 ↔ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
1615adantl 481 . . . . . . . 8 ((𝑑𝐵𝑔 Fn 𝑑) → (∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔 ↔ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
1711, 16mpbird 257 . . . . . . 7 ((𝑑𝐵𝑔 Fn 𝑑) → ∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
189, 17bnj593 34069 . . . . . 6 (𝜑 → ∃𝑑𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
1918bnj937 34095 . . . . 5 (𝜑 → ∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
201, 19bnj835 34083 . . . 4 (𝜓 → ∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
216ssrab3 4080 . . . . . . 7 𝐸𝐷
225bnj1292 34139 . . . . . . 7 𝐷 ⊆ dom 𝑔
2321, 22sstri 3991 . . . . . 6 𝐸 ⊆ dom 𝑔
2423sseli 3978 . . . . 5 (𝑥𝐸𝑥 ∈ dom 𝑔)
251, 24bnj836 34084 . . . 4 (𝜓𝑥 ∈ dom 𝑔)
2620, 25bnj1294 34141 . . 3 (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
272, 3, 4, 5, 6, 7, 1bnj1259 34340 . . . . . . . 8 (𝜑 → ∃𝑑𝐵 Fn 𝑑)
2827bnj1196 34118 . . . . . . 7 (𝜑 → ∃𝑑(𝑑𝐵 Fn 𝑑))
2910adantr 480 . . . . . . . 8 ((𝑑𝐵 Fn 𝑑) → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
30 fndm 6652 . . . . . . . . . 10 ( Fn 𝑑 → dom = 𝑑)
31 sseq2 4008 . . . . . . . . . . 11 (dom = 𝑑 → ( pred(𝑥, 𝐴, 𝑅) ⊆ dom ↔ pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
3231raleqbi1dv 3332 . . . . . . . . . 10 (dom = 𝑑 → (∀𝑥 ∈ dom pred(𝑥, 𝐴, 𝑅) ⊆ dom ↔ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
3330, 32syl 17 . . . . . . . . 9 ( Fn 𝑑 → (∀𝑥 ∈ dom pred(𝑥, 𝐴, 𝑅) ⊆ dom ↔ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
3433adantl 481 . . . . . . . 8 ((𝑑𝐵 Fn 𝑑) → (∀𝑥 ∈ dom pred(𝑥, 𝐴, 𝑅) ⊆ dom ↔ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
3529, 34mpbird 257 . . . . . . 7 ((𝑑𝐵 Fn 𝑑) → ∀𝑥 ∈ dom pred(𝑥, 𝐴, 𝑅) ⊆ dom )
3628, 35bnj593 34069 . . . . . 6 (𝜑 → ∃𝑑𝑥 ∈ dom pred(𝑥, 𝐴, 𝑅) ⊆ dom )
3736bnj937 34095 . . . . 5 (𝜑 → ∀𝑥 ∈ dom pred(𝑥, 𝐴, 𝑅) ⊆ dom )
381, 37bnj835 34083 . . . 4 (𝜓 → ∀𝑥 ∈ dom pred(𝑥, 𝐴, 𝑅) ⊆ dom )
395bnj1293 34140 . . . . . . 7 𝐷 ⊆ dom
4021, 39sstri 3991 . . . . . 6 𝐸 ⊆ dom
4140sseli 3978 . . . . 5 (𝑥𝐸𝑥 ∈ dom )
421, 41bnj836 34084 . . . 4 (𝜓𝑥 ∈ dom )
4338, 42bnj1294 34141 . . 3 (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ dom )
4426, 43ssind 4232 . 2 (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ (dom 𝑔 ∩ dom ))
4544, 5sseqtrrdi 4033 1 (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  {cab 2708  wne 2939  wral 3060  wrex 3069  {crab 3431  cin 3947  wss 3948  cop 4634   class class class wbr 5148  dom cdm 5676  cres 5678   Fn wfn 6538  cfv 6543  w-bnj17 34010   predc-bnj14 34012   FrSe w-bnj15 34016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-bnj17 34011
This theorem is referenced by:  bnj1280  34344
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