Theorem bnj545 34907

Description: Technical lemma for bnj852 34933. 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 1145	. . . . . . . . 9 ⊢ (𝜏 → 𝑓 Fn 𝑚)
3		bnj545.5	. . . . . . . . . 10 ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
4	3	simp1bi 1145	. . . . . . . . 9 ⊢ (𝜎 → 𝑚 ∈ 𝐷)
5	2, 4	anim12i 613	. . . . . . . 8 ⊢ ((𝜏 ∧ 𝜎) → (𝑓 Fn 𝑚 ∧ 𝑚 ∈ 𝐷))
6	5	3adant1 1130	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝑓 Fn 𝑚 ∧ 𝑚 ∈ 𝐷))
7		bnj545.2	. . . . . . . . 9 ⊢ 𝐷 = (ω ∖ {∅})
8	7	bnj529 34753	. . . . . . . 8 ⊢ (𝑚 ∈ 𝐷 → ∅ ∈ 𝑚)
9		fndm 6584	. . . . . . . 8 ⊢ (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
10		eleq2 2820	. . . . . . . . 9 ⊢ (dom 𝑓 = 𝑚 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑚))
11	10	biimparc 479	. . . . . . . 8 ⊢ ((∅ ∈ 𝑚 ∧ dom 𝑓 = 𝑚) → ∅ ∈ dom 𝑓)
12	8, 9, 11	syl2anr 597	. . . . . . 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 6582	. . . . . 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 34782	. . . . 5 ⊢ 𝑓 ⊆ 𝐺
19	16, 18	jctil 519	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝑓 ⊆ 𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
20		df-3an 1088	. . . . 5 ⊢ ((∅ ∈ dom 𝑓 ∧ Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ↔ ((∅ ∈ dom 𝑓 ∧ Fun 𝐺) ∧ 𝑓 ⊆ 𝐺))
21		3anrot 1099	. . . . 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 301	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ ∅ ∈ dom 𝑓) ↔ (𝑓 ⊆ 𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
24	19, 23	sylibr 234	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ ∅ ∈ dom 𝑓))
25		funssfv 6843	. . 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 1146	. . 3 ⊢ (𝜏 → 𝜑′)
28	27	3ad2ant2 1134	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝜑′)
29		bnj545.1	. . . 4 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
30		eqtr 2751	. . . 4 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
31	29, 30	sylan2b 594	. . 3 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
32		bnj545.7	. . 3 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
33	31, 32	sylibr 234	. 2 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → 𝜑″)
34	26, 28, 33	syl2anc 584	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝜑″)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ∖ cdif 3894   ∪ cun 3895   ⊆ wss 3897  ∅c0 4280  {csn 4573  ⟨cop 4579  ∪ ciun 4939  dom cdm 5614  suc csuc 6308  Fun wfun 6475   Fn wfn 6476  ‘cfv 6481  ωcom 7796   predc-bnj14 34700   FrSe w-bnj15 34704

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-res 5626  df-ord 6309  df-on 6310  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489  df-om 7797

This theorem is referenced by:  bnj600  34931  bnj908  34943