Theorem bnj545 32775

Description: Technical lemma for bnj852 32801. 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 1143	. . . . . . . . 9 ⊢ (𝜏 → 𝑓 Fn 𝑚)
3		bnj545.5	. . . . . . . . . 10 ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
4	3	simp1bi 1143	. . . . . . . . 9 ⊢ (𝜎 → 𝑚 ∈ 𝐷)
5	2, 4	anim12i 612	. . . . . . . 8 ⊢ ((𝜏 ∧ 𝜎) → (𝑓 Fn 𝑚 ∧ 𝑚 ∈ 𝐷))
6	5	3adant1 1128	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝑓 Fn 𝑚 ∧ 𝑚 ∈ 𝐷))
7		bnj545.2	. . . . . . . . 9 ⊢ 𝐷 = (ω ∖ {∅})
8	7	bnj529 32621	. . . . . . . 8 ⊢ (𝑚 ∈ 𝐷 → ∅ ∈ 𝑚)
9		fndm 6520	. . . . . . . 8 ⊢ (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
10		eleq2 2827	. . . . . . . . 9 ⊢ (dom 𝑓 = 𝑚 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑚))
11	10	biimparc 479	. . . . . . . 8 ⊢ ((∅ ∈ 𝑚 ∧ dom 𝑓 = 𝑚) → ∅ ∈ dom 𝑓)
12	8, 9, 11	syl2anr 596	. . . . . . 7 ⊢ ((𝑓 Fn 𝑚 ∧ 𝑚 ∈ 𝐷) → ∅ ∈ dom 𝑓)
13	6, 12	syl 17	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → ∅ ∈ dom 𝑓)
14		bnj545.6	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)
15	14	fnfund 6518	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → Fun 𝐺)
16	13, 15	jca 511	. . . . 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 32650	. . . . 5 ⊢ 𝑓 ⊆ 𝐺
19	16, 18	jctil 519	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝑓 ⊆ 𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
20		df-3an 1087	. . . . 5 ⊢ ((∅ ∈ dom 𝑓 ∧ Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ↔ ((∅ ∈ dom 𝑓 ∧ Fun 𝐺) ∧ 𝑓 ⊆ 𝐺))
21		3anrot 1098	. . . . 5 ⊢ ((∅ ∈ dom 𝑓 ∧ Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ↔ (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ ∅ ∈ dom 𝑓))
22		ancom 460	. . . . 5 ⊢ (((∅ ∈ dom 𝑓 ∧ Fun 𝐺) ∧ 𝑓 ⊆ 𝐺) ↔ (𝑓 ⊆ 𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
23	20, 21, 22	3bitr3i 300	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ ∅ ∈ dom 𝑓) ↔ (𝑓 ⊆ 𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
24	19, 23	sylibr 233	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ ∅ ∈ dom 𝑓))
25		funssfv 6777	. . 3 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ ∅ ∈ dom 𝑓) → (𝐺‘∅) = (𝑓‘∅))
26	24, 25	syl 17	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝐺‘∅) = (𝑓‘∅))
27	1	simp2bi 1144	. . 3 ⊢ (𝜏 → 𝜑′)
28	27	3ad2ant2 1132	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝜑′)
29		bnj545.1	. . . 4 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
30		eqtr 2761	. . . 4 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
31	29, 30	sylan2b 593	. . 3 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
32		bnj545.7	. . 3 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
33	31, 32	sylibr 233	. 2 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → 𝜑″)
34	26, 28, 33	syl2anc 583	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝜑″)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  {csn 4558  ⟨cop 4564  ∪ ciun 4921  dom cdm 5580  suc csuc 6253  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418  ωcom 7687   predc-bnj14 32567   FrSe w-bnj15 32571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-ord 6254  df-on 6255  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426  df-om 7688

This theorem is referenced by:  bnj600  32799  bnj908  32811