Theorem bnj545 35525

Description: Technical lemma for bnj852 35551. 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
bnj545.1	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj545.2	⊢ 𝐷 = (ω ∖ {∅})
bnj545.3	⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj545.4	⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
bnj545.5	⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
bnj545.6	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)
bnj545.7	⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))

Assertion

Ref	Expression
bnj545	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝜑″)

Proof of Theorem bnj545

Step	Hyp	Ref	Expression
1		bnj545.4	. . . . . . . . . 10 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
2	1	simp1bi 1163	. . . . . . . . 9 ⊢ (𝜏 → 𝑓 Fn 𝑚)
3		bnj545.5	. . . . . . . . . 10 ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
4	3	simp1bi 1163	. . . . . . . . 9 ⊢ (𝜎 → 𝑚 ∈ 𝐷)
5	2, 4	anim12i 625	. . . . . . . 8 ⊢ ((𝜏 ∧ 𝜎) → (𝑓 Fn 𝑚 ∧ 𝑚 ∈ 𝐷))
6	5	3adant1 1148	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝑓 Fn 𝑚 ∧ 𝑚 ∈ 𝐷))
7		bnj545.2	. . . . . . . . 9 ⊢ 𝐷 = (ω ∖ {∅})
8	7	bnj529 35372	. . . . . . . 8 ⊢ (𝑚 ∈ 𝐷 → ∅ ∈ 𝑚)
9		fndm 6642	. . . . . . . 8 ⊢ (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
10		eleq2 2849	. . . . . . . . 9 ⊢ (dom 𝑓 = 𝑚 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑚))
11	10	biimparc 485	. . . . . . . 8 ⊢ ((∅ ∈ 𝑚 ∧ dom 𝑓 = 𝑚) → ∅ ∈ dom 𝑓)
12	8, 9, 11	syl2anr 609	. . . . . . 7 ⊢ ((𝑓 Fn 𝑚 ∧ 𝑚 ∈ 𝐷) → ∅ ∈ dom 𝑓)
13	6, 12	syl 18	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → ∅ ∈ dom 𝑓)
14		bnj545.6	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)
15	14	fnfund 6640	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → Fun 𝐺)
16	13, 15	jca 521	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (∅ ∈ dom 𝑓 ∧ Fun 𝐺))
17		bnj545.3	. . . . . 6 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
18	17	bnj931 35401	. . . . 5 ⊢ 𝑓 ⊆ 𝐺
19	16, 18	jctil 529	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝑓 ⊆ 𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
20		df-3an 1105	. . . . 5 ⊢ ((∅ ∈ dom 𝑓 ∧ Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ↔ ((∅ ∈ dom 𝑓 ∧ Fun 𝐺) ∧ 𝑓 ⊆ 𝐺))
21		3anrot 1117	. . . . 5 ⊢ ((∅ ∈ dom 𝑓 ∧ Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ↔ (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ ∅ ∈ dom 𝑓))
22		ancom 466	. . . . 5 ⊢ (((∅ ∈ dom 𝑓 ∧ Fun 𝐺) ∧ 𝑓 ⊆ 𝐺) ↔ (𝑓 ⊆ 𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
23	20, 21, 22	3bitr3i 304	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ ∅ ∈ dom 𝑓) ↔ (𝑓 ⊆ 𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
24	19, 23	sylibr 237	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ ∅ ∈ dom 𝑓))
25		funssfv 6906	. . 3 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ ∅ ∈ dom 𝑓) → (𝐺‘∅) = (𝑓‘∅))
26	24, 25	syl 18	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝐺‘∅) = (𝑓‘∅))
27	1	simp2bi 1164	. . 3 ⊢ (𝜏 → 𝜑′)
28	27	3ad2ant2 1152	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝜑′)
29		bnj545.1	. . . 4 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
30		eqtr 2780	. . . 4 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
31	29, 30	sylan2b 606	. . 3 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
32		bnj545.7	. . 3 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
33	31, 32	sylibr 237	. 2 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → 𝜑″)
34	26, 28, 33	syl2anc 596	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝜑″)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∖ cdif 4045   ∪ cun 4149  ∅c0 4278  {csn 4583  ⟨cop 4589  ∪ ciun 4950  dom cdm 5650  suc csuc 6364  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538  ωcom 7877   predc-bnj14 35319   FrSe w-bnj15 35323

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-res 5662  df-ord 6365  df-on 6366  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546  df-om 7878

This theorem is used by:  bnj600  35549  bnj908  35561