Theorem bnj1286 35016

Description: Technical lemma for bnj60 35059. 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
bnj1286.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1286.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1286.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1286.4	⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)
bnj1286.5	⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}
bnj1286.6	⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))
bnj1286.7	⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))

Assertion

Ref	Expression
bnj1286	⊢ (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷)

Distinct variable groups:   𝐴,𝑑,𝑓   𝐵,𝑓,𝑔   𝐵,ℎ,𝑓   𝑥,𝐷   𝑓,𝐺,𝑔   ℎ,𝐺   𝑅,𝑑,𝑓   𝑔,𝑌   ℎ,𝑌   𝑔,𝑑,𝑥,𝑓   ℎ,𝑑,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝜓(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝐴(𝑥,𝑦,𝑔,ℎ)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝐷(𝑦,𝑓,𝑔,ℎ,𝑑)   𝑅(𝑥,𝑦,𝑔,ℎ)   𝐸(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝐺(𝑥,𝑦,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1286

Step	Hyp	Ref	Expression
1		bnj1286.7	. . . . 5 ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))
2		bnj1286.1	. . . . . . . . 9 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
3		bnj1286.2	. . . . . . . . 9 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
4		bnj1286.3	. . . . . . . . 9 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
5		bnj1286.4	. . . . . . . . 9 ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)
6		bnj1286.5	. . . . . . . . 9 ⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}
7		bnj1286.6	. . . . . . . . 9 ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))
8	2, 3, 4, 5, 6, 7, 1	bnj1256 35012	. . . . . . . 8 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑)
9	8	bnj1196 34791	. . . . . . 7 ⊢ (𝜑 → ∃𝑑(𝑑 ∈ 𝐵 ∧ 𝑔 Fn 𝑑))
10	2	bnj1517 34847	. . . . . . . . 9 ⊢ (𝑑 ∈ 𝐵 → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝑑 ∈ 𝐵 ∧ 𝑔 Fn 𝑑) → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
12		fndm 6624	. . . . . . . . . 10 ⊢ (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
13		sseq2 3976	. . . . . . . . . . 11 ⊢ (dom 𝑔 = 𝑑 → ( pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔 ↔ pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
14	13	raleqbi1dv 3313	. . . . . . . . . 10 ⊢ (dom 𝑔 = 𝑑 → (∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔 ↔ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
15	12, 14	syl 17	. . . . . . . . 9 ⊢ (𝑔 Fn 𝑑 → (∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔 ↔ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
16	15	adantl 481	. . . . . . . 8 ⊢ ((𝑑 ∈ 𝐵 ∧ 𝑔 Fn 𝑑) → (∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔 ↔ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
17	11, 16	mpbird 257	. . . . . . 7 ⊢ ((𝑑 ∈ 𝐵 ∧ 𝑔 Fn 𝑑) → ∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
18	9, 17	bnj593 34742	. . . . . 6 ⊢ (𝜑 → ∃𝑑∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
19	18	bnj937 34768	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
20	1, 19	bnj835 34756	. . . 4 ⊢ (𝜓 → ∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
21	6	ssrab3 4048	. . . . . . 7 ⊢ 𝐸 ⊆ 𝐷
22	5	bnj1292 34812	. . . . . . 7 ⊢ 𝐷 ⊆ dom 𝑔
23	21, 22	sstri 3959	. . . . . 6 ⊢ 𝐸 ⊆ dom 𝑔
24	23	sseli 3945	. . . . 5 ⊢ (𝑥 ∈ 𝐸 → 𝑥 ∈ dom 𝑔)
25	1, 24	bnj836 34757	. . . 4 ⊢ (𝜓 → 𝑥 ∈ dom 𝑔)
26	20, 25	bnj1294 34814	. . 3 ⊢ (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
27	2, 3, 4, 5, 6, 7, 1	bnj1259 35013	. . . . . . . 8 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑)
28	27	bnj1196 34791	. . . . . . 7 ⊢ (𝜑 → ∃𝑑(𝑑 ∈ 𝐵 ∧ ℎ Fn 𝑑))
29	10	adantr 480	. . . . . . . 8 ⊢ ((𝑑 ∈ 𝐵 ∧ ℎ Fn 𝑑) → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
30		fndm 6624	. . . . . . . . . 10 ⊢ (ℎ Fn 𝑑 → dom ℎ = 𝑑)
31		sseq2 3976	. . . . . . . . . . 11 ⊢ (dom ℎ = 𝑑 → ( pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ ↔ pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
32	31	raleqbi1dv 3313	. . . . . . . . . 10 ⊢ (dom ℎ = 𝑑 → (∀𝑥 ∈ dom ℎ pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ ↔ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
33	30, 32	syl 17	. . . . . . . . 9 ⊢ (ℎ Fn 𝑑 → (∀𝑥 ∈ dom ℎ pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ ↔ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
34	33	adantl 481	. . . . . . . 8 ⊢ ((𝑑 ∈ 𝐵 ∧ ℎ Fn 𝑑) → (∀𝑥 ∈ dom ℎ pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ ↔ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
35	29, 34	mpbird 257	. . . . . . 7 ⊢ ((𝑑 ∈ 𝐵 ∧ ℎ Fn 𝑑) → ∀𝑥 ∈ dom ℎ pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ)
36	28, 35	bnj593 34742	. . . . . 6 ⊢ (𝜑 → ∃𝑑∀𝑥 ∈ dom ℎ pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ)
37	36	bnj937 34768	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ dom ℎ pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ)
38	1, 37	bnj835 34756	. . . 4 ⊢ (𝜓 → ∀𝑥 ∈ dom ℎ pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ)
39	5	bnj1293 34813	. . . . . . 7 ⊢ 𝐷 ⊆ dom ℎ
40	21, 39	sstri 3959	. . . . . 6 ⊢ 𝐸 ⊆ dom ℎ
41	40	sseli 3945	. . . . 5 ⊢ (𝑥 ∈ 𝐸 → 𝑥 ∈ dom ℎ)
42	1, 41	bnj836 34757	. . . 4 ⊢ (𝜓 → 𝑥 ∈ dom ℎ)
43	38, 42	bnj1294 34814	. . 3 ⊢ (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ)
44	26, 43	ssind 4207	. 2 ⊢ (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ (dom 𝑔 ∩ dom ℎ))
45	44, 5	sseqtrrdi 3991	1 ⊢ (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2708   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  {crab 3408   ∩ cin 3916   ⊆ wss 3917  ⟨cop 4598   class class class wbr 5110  dom cdm 5641   ↾ cres 5643   Fn wfn 6509  ‘cfv 6514   ∧ w-bnj17 34683   predc-bnj14 34685   FrSe w-bnj15 34689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522  df-bnj17 34684

This theorem is referenced by:  bnj1280  35017