Theorem bnj999 35644

Description: Technical lemma for bnj69 35696. 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 3458	. . . . . . 7 ⊢ 𝑝 ∈ V
6	1, 2, 3, 4, 5	bnj919 35454	. . . . . 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 35453	. . . . . 6 ⊢ 𝐺 ∈ V
12	6, 7, 8, 9, 11	bnj976 35464	. . . . 5 ⊢ (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″))
13	12	bnj1254 35495	. . . 4 ⊢ (𝜒″ → 𝜓″)
14	13	anim1i 627	. . 3 ⊢ ((𝜒″ ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖))) → (𝜓″ ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖))))
15		bnj252 35390	. . 3 ⊢ ((𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) ↔ (𝜒″ ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖))))
16		bnj252 35390	. . 3 ⊢ ((𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) ↔ (𝜓″ ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖))))
17	14, 15, 16	3imtr4i 295	. 2 ⊢ ((𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) → (𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)))
18		ssiun2 5009	. . . 4 ⊢ (𝑦 ∈ (𝐺‘𝑖) → pred(𝑦, 𝐴, 𝑅) ⊆ ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
19	18	bnj708 35443	. . 3 ⊢ ((𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
20		3simpa 1166	. . . . . 6 ⊢ ((𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) → (𝜓″ ∧ 𝑖 ∈ ω))
21	20	ancomd 467	. . . . 5 ⊢ ((𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) → (𝑖 ∈ ω ∧ 𝜓″))
22		simp3 1156	. . . . 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 35577	. . . . . . 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 35628	. . . . . 6 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑝 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
27	26	bnj228 35422	. . . . 5 ⊢ ((𝑖 ∈ ω ∧ 𝜓″) → (suc 𝑖 ∈ 𝑝 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
28	21, 22, 27	sylc 66	. . . 4 ⊢ ((𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
29	28	bnj721 35444	. . 3 ⊢ ((𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
30	19, 29	sseqtrrd 3950	. 2 ⊢ ((𝜓″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
31	17, 30	syl 18	1 ⊢ ((𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∀wral 3080  [wsbc 3742   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  {csn 4588  ⟨cop 4838  ∪ ciun 4954  suc csuc 6371   Fn wfn 6542  ‘cfv 6547  ωcom 7887   ∧ w-bnj17 35373   predc-bnj14 35375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6501  df-fun 6549  df-fn 6550  df-fv 6555  df-bnj17 35374

This theorem is used by:  bnj1006  35646