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Theorem bnj999 30735
Description: Technical lemma for bnj69 30786. 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 3189 . . . . . . 7 𝑝 ∈ V
61, 2, 3, 4, 5bnj919 30545 . . . . . 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 30544 . . . . . 6 𝐺 ∈ V
126, 7, 8, 9, 11bnj976 30556 . . . . 5 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
1312bnj1254 30588 . . . 4 (𝜒″𝜓″)
1413anim1i 591 . . 3 ((𝜒″ ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖))) → (𝜓″ ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖))))
15 bnj252 30476 . . 3 ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) ↔ (𝜒″ ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖))))
16 bnj252 30476 . . 3 ((𝜓″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) ↔ (𝜓″ ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖))))
1714, 15, 163imtr4i 281 . 2 ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) → (𝜓″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)))
18 ssiun2 4529 . . . 4 (𝑦 ∈ (𝐺𝑖) → pred(𝑦, 𝐴, 𝑅) ⊆ 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
1918bnj708 30534 . . 3 ((𝜓″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
20 3simpa 1056 . . . . . 6 ((𝜓″𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝜓″𝑖 ∈ ω))
2120ancomd 467 . . . . 5 ((𝜓″𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝑖 ∈ ω ∧ 𝜓″))
22 simp3 1061 . . . . 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 30669 . . . . . . 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 30720 . . . . . 6 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑝 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
2726bnj228 30511 . . . . 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 30535 . . 3 ((𝜓″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
3019, 29sseqtr4d 3621 . 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  [wsbc 3417  cun 3553  wss 3555  c0 3891  {csn 4148  cop 4154   ciun 4485  suc csuc 5684   Fn wfn 5842  cfv 5847  ωcom 7012  w-bnj17 30459   predc-bnj14 30461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-bnj17 30460
This theorem is referenced by:  bnj1006  30737
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