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Theorem bnj1286 31425
Description: Technical lemma for bnj60 31468. 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 31421 . . . . . . . 8 (𝜑 → ∃𝑑𝐵 𝑔 Fn 𝑑)
98bnj1196 31203 . . . . . . 7 (𝜑 → ∃𝑑(𝑑𝐵𝑔 Fn 𝑑))
102bnj1517 31258 . . . . . . . . 9 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
1110adantr 466 . . . . . . . 8 ((𝑑𝐵𝑔 Fn 𝑑) → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
12 fndm 6130 . . . . . . . . . 10 (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
13 sseq2 3776 . . . . . . . . . . 11 (dom 𝑔 = 𝑑 → ( pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔 ↔ pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
1413raleqbi1dv 3295 . . . . . . . . . 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 467 . . . . . . . 8 ((𝑑𝐵𝑔 Fn 𝑑) → (∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔 ↔ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
1711, 16mpbird 247 . . . . . . 7 ((𝑑𝐵𝑔 Fn 𝑑) → ∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
189, 17bnj593 31153 . . . . . 6 (𝜑 → ∃𝑑𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
1918bnj937 31180 . . . . 5 (𝜑 → ∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
201, 19bnj835 31167 . . . 4 (𝜓 → ∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
216ssrab3 3837 . . . . . . 7 𝐸𝐷
225bnj1292 31224 . . . . . . 7 𝐷 ⊆ dom 𝑔
2321, 22sstri 3761 . . . . . 6 𝐸 ⊆ dom 𝑔
2423sseli 3748 . . . . 5 (𝑥𝐸𝑥 ∈ dom 𝑔)
251, 24bnj836 31168 . . . 4 (𝜓𝑥 ∈ dom 𝑔)
2620, 25bnj1294 31226 . . 3 (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
272, 3, 4, 5, 6, 7, 1bnj1259 31422 . . . . . . . 8 (𝜑 → ∃𝑑𝐵 Fn 𝑑)
2827bnj1196 31203 . . . . . . 7 (𝜑 → ∃𝑑(𝑑𝐵 Fn 𝑑))
2910adantr 466 . . . . . . . 8 ((𝑑𝐵 Fn 𝑑) → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
30 fndm 6130 . . . . . . . . . 10 ( Fn 𝑑 → dom = 𝑑)
31 sseq2 3776 . . . . . . . . . . 11 (dom = 𝑑 → ( pred(𝑥, 𝐴, 𝑅) ⊆ dom ↔ pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
3231raleqbi1dv 3295 . . . . . . . . . 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 467 . . . . . . . 8 ((𝑑𝐵 Fn 𝑑) → (∀𝑥 ∈ dom pred(𝑥, 𝐴, 𝑅) ⊆ dom ↔ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
3529, 34mpbird 247 . . . . . . 7 ((𝑑𝐵 Fn 𝑑) → ∀𝑥 ∈ dom pred(𝑥, 𝐴, 𝑅) ⊆ dom )
3628, 35bnj593 31153 . . . . . 6 (𝜑 → ∃𝑑𝑥 ∈ dom pred(𝑥, 𝐴, 𝑅) ⊆ dom )
3736bnj937 31180 . . . . 5 (𝜑 → ∀𝑥 ∈ dom pred(𝑥, 𝐴, 𝑅) ⊆ dom )
381, 37bnj835 31167 . . . 4 (𝜓 → ∀𝑥 ∈ dom pred(𝑥, 𝐴, 𝑅) ⊆ dom )
395bnj1293 31225 . . . . . . 7 𝐷 ⊆ dom
4021, 39sstri 3761 . . . . . 6 𝐸 ⊆ dom
4140sseli 3748 . . . . 5 (𝑥𝐸𝑥 ∈ dom )
421, 41bnj836 31168 . . . 4 (𝜓𝑥 ∈ dom )
4338, 42bnj1294 31226 . . 3 (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ dom )
4426, 43ssind 3985 . 2 (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ (dom 𝑔 ∩ dom ))
4544, 5syl6sseqr 3801 1 (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  {cab 2757  wne 2943  wral 3061  wrex 3062  {crab 3065  cin 3722  wss 3723  cop 4322   class class class wbr 4786  dom cdm 5249  cres 5251   Fn wfn 6026  cfv 6031  w-bnj17 31092   predc-bnj14 31094   FrSe w-bnj15 31098
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-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-bnj17 31093
This theorem is referenced by:  bnj1280  31426
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