Theorem bnj999 33073

Description: Technical lemma for bnj69 33125. 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 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj999.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
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 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
bnj999.16	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})

Assertion

Ref	Expression
bnj999	⊢ ((𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) → 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

Step	Hyp	Ref	Expression
1		bnj999.3	. . . . . . 7 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
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 3444	. . . . . . 7 ⊢ 𝑝 ∈ V
6	1, 2, 3, 4, 5	bnj919 32882	. . . . . 6 ⊢ (𝜒′ ↔ (𝑝 ∈ 𝐷 ∧ 𝑓 Fn 𝑝 ∧ 𝜑′ ∧ 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⟩})
11	10	bnj918 32881	. . . . . 6 ⊢ 𝐺 ∈ V
12	6, 7, 8, 9, 11	bnj976 32892	. . . . 5 ⊢ (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″))
13	12	bnj1254 32924	. . . 4 ⊢ (𝜒″ → 𝜓″)
14	13	anim1i 615	. . 3 ⊢ ((𝜒″ ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖))) → (𝜓″ ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖))))
15		bnj252 32818	. . 3 ⊢ ((𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) ↔ (𝜒″ ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖))))
16		bnj252 32818	. . 3 ⊢ ((𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) ↔ (𝜓″ ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖))))
17	14, 15, 16	3imtr4i 291	. 2 ⊢ ((𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) → (𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)))
18		ssiun2 4989	. . . 4 ⊢ (𝑦 ∈ (𝐺‘𝑖) → pred(𝑦, 𝐴, 𝑅) ⊆ ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
19	18	bnj708 32871	. . 3 ⊢ ((𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
20		3simpa 1147	. . . . . 6 ⊢ ((𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) → (𝜓″ ∧ 𝑖 ∈ ω))
21	20	ancomd 462	. . . . 5 ⊢ ((𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) → (𝑖 ∈ ω ∧ 𝜓″))
22		simp3 1137	. . . . 5 ⊢ ((𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) → suc 𝑖 ∈ 𝑝)
23		bnj999.2	. . . . . . . 8 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
24	23, 3, 5	bnj539 33006	. . . . . . 7 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑝 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
25		bnj999.15	. . . . . . 7 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
26	24, 8, 25, 10	bnj965 33057	. . . . . 6 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑝 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
27	26	bnj228 32850	. . . . 5 ⊢ ((𝑖 ∈ ω ∧ 𝜓″) → (suc 𝑖 ∈ 𝑝 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
28	21, 22, 27	sylc 65	. . . 4 ⊢ ((𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
29	28	bnj721 32872	. . 3 ⊢ ((𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
30	19, 29	sseqtrrd 3971	. 2 ⊢ ((𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
31	17, 30	syl 17	1 ⊢ ((𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3061  [wsbc 3725   ∪ cun 3894   ⊆ wss 3896  ∅c0 4266  {csn 4570  ⟨cop 4576  ∪ ciun 4936  suc csuc 6290   Fn wfn 6460  ‘cfv 6465  ωcom 7758   ∧ w-bnj17 32801   predc-bnj14 32803

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-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5237  ax-nul 5244  ax-pr 5366  ax-un 7629

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3442  df-sbc 3726  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-iun 4938  df-br 5087  df-opab 5149  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-iota 6417  df-fun 6467  df-fn 6468  df-fv 6473  df-bnj17 32802

This theorem is referenced by:  bnj1006  33075