Theorem bnj1286 35373

Description: Technical lemma for bnj60 35416. 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 35369	. . . . . . . 8 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑)
9	8	bnj1196 35148	. . . . . . 7 ⊢ (𝜑 → ∃𝑑(𝑑 ∈ 𝐵 ∧ 𝑔 Fn 𝑑))
10	2	bnj1517 35204	. . . . . . . . 9 ⊢ (𝑑 ∈ 𝐵 → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
11	10	adantr 485	. . . . . . . 8 ⊢ ((𝑑 ∈ 𝐵 ∧ 𝑔 Fn 𝑑) → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
12		fndm 6638	. . . . . . . . . 10 ⊢ (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
13		sseq2 3962	. . . . . . . . . . 11 ⊢ (dom 𝑔 = 𝑑 → ( pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔 ↔ pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
14	13	raleqbi1dv 3331	. . . . . . . . . 10 ⊢ (dom 𝑔 = 𝑑 → (∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔 ↔ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
15	12, 14	syl 18	. . . . . . . . 9 ⊢ (𝑔 Fn 𝑑 → (∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔 ↔ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
16	15	adantl 486	. . . . . . . 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 260	. . . . . . 7 ⊢ ((𝑑 ∈ 𝐵 ∧ 𝑔 Fn 𝑑) → ∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
18	9, 17	bnj593 35100	. . . . . 6 ⊢ (𝜑 → ∃𝑑∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
19	18	bnj937 35126	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
20	1, 19	bnj835 35114	. . . 4 ⊢ (𝜓 → ∀𝑥 ∈ dom 𝑔 pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
21	6	ssrab3 4035	. . . . . . 7 ⊢ 𝐸 ⊆ 𝐷
22	5	bnj1292 35169	. . . . . . 7 ⊢ 𝐷 ⊆ dom 𝑔
23	21, 22	sstri 3945	. . . . . 6 ⊢ 𝐸 ⊆ dom 𝑔
24	23	sseli 3932	. . . . 5 ⊢ (𝑥 ∈ 𝐸 → 𝑥 ∈ dom 𝑔)
25	1, 24	bnj836 35115	. . . 4 ⊢ (𝜓 → 𝑥 ∈ dom 𝑔)
26	20, 25	bnj1294 35171	. . 3 ⊢ (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑔)
27	2, 3, 4, 5, 6, 7, 1	bnj1259 35370	. . . . . . . 8 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑)
28	27	bnj1196 35148	. . . . . . 7 ⊢ (𝜑 → ∃𝑑(𝑑 ∈ 𝐵 ∧ ℎ Fn 𝑑))
29	10	adantr 485	. . . . . . . 8 ⊢ ((𝑑 ∈ 𝐵 ∧ ℎ Fn 𝑑) → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
30		fndm 6638	. . . . . . . . . 10 ⊢ (ℎ Fn 𝑑 → dom ℎ = 𝑑)
31		sseq2 3962	. . . . . . . . . . 11 ⊢ (dom ℎ = 𝑑 → ( pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ ↔ pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
32	31	raleqbi1dv 3331	. . . . . . . . . 10 ⊢ (dom ℎ = 𝑑 → (∀𝑥 ∈ dom ℎ pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ ↔ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
33	30, 32	syl 18	. . . . . . . . 9 ⊢ (ℎ Fn 𝑑 → (∀𝑥 ∈ dom ℎ pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ ↔ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
34	33	adantl 486	. . . . . . . 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 260	. . . . . . 7 ⊢ ((𝑑 ∈ 𝐵 ∧ ℎ Fn 𝑑) → ∀𝑥 ∈ dom ℎ pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ)
36	28, 35	bnj593 35100	. . . . . 6 ⊢ (𝜑 → ∃𝑑∀𝑥 ∈ dom ℎ pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ)
37	36	bnj937 35126	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ dom ℎ pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ)
38	1, 37	bnj835 35114	. . . 4 ⊢ (𝜓 → ∀𝑥 ∈ dom ℎ pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ)
39	5	bnj1293 35170	. . . . . . 7 ⊢ 𝐷 ⊆ dom ℎ
40	21, 39	sstri 3945	. . . . . 6 ⊢ 𝐸 ⊆ dom ℎ
41	40	sseli 3932	. . . . 5 ⊢ (𝑥 ∈ 𝐸 → 𝑥 ∈ dom ℎ)
42	1, 41	bnj836 35115	. . . 4 ⊢ (𝜓 → 𝑥 ∈ dom ℎ)
43	38, 42	bnj1294 35171	. . 3 ⊢ (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ dom ℎ)
44	26, 43	ssind 4192	. 2 ⊢ (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ (dom 𝑔 ∩ dom ℎ))
45	44, 5	sseqtrrdi 3977	1 ⊢ (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  {cab 2739   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  {crab 3414   ∩ cin 3903   ⊆ wss 3904  ⟨cop 4594   class class class wbr 5108  dom cdm 5661   ↾ cres 5663   Fn wfn 6531  ‘cfv 6536   ∧ w-bnj17 35041   predc-bnj14 35043   FrSe w-bnj15 35047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-bnj17 35042

This theorem is referenced by:  bnj1280  35374