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Theorem bnj999 31845
Description: Technical lemma for bnj69 31896. 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
bnj999.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj999.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj999.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj999.7 (𝜑′[𝑝 / 𝑛]𝜑)
bnj999.8 (𝜓′[𝑝 / 𝑛]𝜓)
bnj999.9 (𝜒′[𝑝 / 𝑛]𝜒)
bnj999.10 (𝜑″[𝐺 / 𝑓]𝜑′)
bnj999.11 (𝜓″[𝐺 / 𝑓]𝜓′)
bnj999.12 (𝜒″[𝐺 / 𝑓]𝜒′)
bnj999.15 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj999.16 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
Assertion
Ref Expression
bnj999 ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
Distinct variable groups:   𝑓,𝑖,𝑛,𝑦   𝐴,𝑓,𝑛   𝐷,𝑓,𝑛   𝑖,𝐺   𝑅,𝑓,𝑛   𝑓,𝑋,𝑛   𝑓,𝑝,𝑖,𝑛
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑦,𝑖,𝑚,𝑝)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑖,𝑚,𝑝)   𝑅(𝑦,𝑖,𝑚,𝑝)   𝐺(𝑦,𝑓,𝑚,𝑛,𝑝)   𝑋(𝑦,𝑖,𝑚,𝑝)   𝜑′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑″(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒″(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj999
StepHypRef Expression
1 bnj999.3 . . . . . . 7 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
2 bnj999.7 . . . . . . 7 (𝜑′[𝑝 / 𝑛]𝜑)
3 bnj999.8 . . . . . . 7 (𝜓′[𝑝 / 𝑛]𝜓)
4 bnj999.9 . . . . . . 7 (𝜒′[𝑝 / 𝑛]𝜒)
5 vex 3440 . . . . . . 7 𝑝 ∈ V
61, 2, 3, 4, 5bnj919 31655 . . . . . 6 (𝜒′ ↔ (𝑝𝐷𝑓 Fn 𝑝𝜑′𝜓′))
7 bnj999.10 . . . . . 6 (𝜑″[𝐺 / 𝑓]𝜑′)
8 bnj999.11 . . . . . 6 (𝜓″[𝐺 / 𝑓]𝜓′)
9 bnj999.12 . . . . . 6 (𝜒″[𝐺 / 𝑓]𝜒′)
10 bnj999.16 . . . . . . 7 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
1110bnj918 31654 . . . . . 6 𝐺 ∈ V
126, 7, 8, 9, 11bnj976 31666 . . . . 5 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
1312bnj1254 31698 . . . 4 (𝜒″𝜓″)
1413anim1i 614 . . 3 ((𝜒″ ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖))) → (𝜓″ ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖))))
15 bnj252 31590 . . 3 ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) ↔ (𝜒″ ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖))))
16 bnj252 31590 . . 3 ((𝜓″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) ↔ (𝜓″ ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖))))
1714, 15, 163imtr4i 293 . 2 ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) → (𝜓″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)))
18 ssiun2 4870 . . . 4 (𝑦 ∈ (𝐺𝑖) → pred(𝑦, 𝐴, 𝑅) ⊆ 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
1918bnj708 31644 . . 3 ((𝜓″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
20 3simpa 1141 . . . . . 6 ((𝜓″𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝜓″𝑖 ∈ ω))
2120ancomd 462 . . . . 5 ((𝜓″𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝑖 ∈ ω ∧ 𝜓″))
22 simp3 1131 . . . . 5 ((𝜓″𝑖 ∈ ω ∧ suc 𝑖𝑝) → suc 𝑖𝑝)
23 bnj999.2 . . . . . . . 8 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
2423, 3, 5bnj539 31779 . . . . . . 7 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑝 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
25 bnj999.15 . . . . . . 7 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
2624, 8, 25, 10bnj965 31830 . . . . . 6 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑝 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
2726bnj228 31622 . . . . 5 ((𝑖 ∈ ω ∧ 𝜓″) → (suc 𝑖𝑝 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
2821, 22, 27sylc 65 . . . 4 ((𝜓″𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
2928bnj721 31645 . . 3 ((𝜓″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
3019, 29sseqtr4d 3929 . 2 ((𝜓″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
3117, 30syl 17 1 ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wral 3105  [wsbc 3706  cun 3857  wss 3859  c0 4211  {csn 4472  cop 4478   ciun 4825  suc csuc 6068   Fn wfn 6220  cfv 6225  ωcom 7436  w-bnj17 31573   predc-bnj14 31575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-iota 6189  df-fun 6227  df-fn 6228  df-fv 6233  df-bnj17 31574
This theorem is referenced by:  bnj1006  31847
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