Theorem mreexexlem3d 16306

Description: Base case of the induction in mreexexd 16308. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
mreexexlem2d.1	⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
mreexexlem2d.2	⊢ 𝑁 = (mrCls‘𝐴)
mreexexlem2d.3	⊢ 𝐼 = (mrInd‘𝐴)
mreexexlem2d.4	⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
mreexexlem2d.5	⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻))
mreexexlem2d.6	⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻))
mreexexlem2d.7	⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
mreexexlem2d.8	⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼)
mreexexlem3d.9	⊢ (𝜑 → (𝐹 = ∅ ∨ 𝐺 = ∅))

Assertion

Ref	Expression
mreexexlem3d	⊢ (𝜑 → ∃𝑖 ∈ 𝒫 𝐺(𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼))

Distinct variable groups:   𝑖,𝐹   𝑖,𝐺   𝑖,𝐻   𝑖,𝐼

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑖,𝑠)   𝐴(𝑦,𝑧,𝑖,𝑠)   𝐹(𝑦,𝑧,𝑠)   𝐺(𝑦,𝑧,𝑠)   𝐻(𝑦,𝑧,𝑠)   𝐼(𝑦,𝑧,𝑠)   𝑁(𝑦,𝑧,𝑖,𝑠)   𝑋(𝑦,𝑧,𝑖,𝑠)

Proof of Theorem mreexexlem3d

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = ∅) → 𝐹 = ∅)
2		mreexexlem2d.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
3	2	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐴 ∈ (Moore‘𝑋))
4		mreexexlem2d.2	. . . . . . . . 9 ⊢ 𝑁 = (mrCls‘𝐴)
5		mreexexlem2d.3	. . . . . . . . 9 ⊢ 𝐼 = (mrInd‘𝐴)
6		mreexexlem2d.7	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
7	6	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
8		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐺 = ∅)
9	8	uneq1d 3766	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐺 ∪ 𝐻) = (∅ ∪ 𝐻))
10		uncom 3757	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∪ ∅) = (∅ ∪ 𝐻)
11		un0 3967	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∪ ∅) = 𝐻
12	10, 11	eqtr3i 2646	. . . . . . . . . . . . 13 ⊢ (∅ ∪ 𝐻) = 𝐻
13	9, 12	syl6eq 2672	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐺 ∪ 𝐻) = 𝐻)
14	13	fveq2d 6195	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝑁‘(𝐺 ∪ 𝐻)) = (𝑁‘𝐻))
15	7, 14	sseqtrd 3641	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝑁‘𝐻))
16		mreexexlem2d.8	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼)
17	16	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) ∈ 𝐼)
18	5, 3, 17	mrissd 16296	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) ⊆ 𝑋)
19	18	unssbd 3791	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐻 ⊆ 𝑋)
20	3, 4, 19	mrcssidd 16285	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐻 ⊆ (𝑁‘𝐻))
21	15, 20	unssd 3789	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) ⊆ (𝑁‘𝐻))
22		ssun2 3777	. . . . . . . . . 10 ⊢ 𝐻 ⊆ (𝐹 ∪ 𝐻)
23	22	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐻 ⊆ (𝐹 ∪ 𝐻))
24	3, 4, 5, 21, 23, 17	mrissmrcd 16300	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) = 𝐻)
25		ssequn1 3783	. . . . . . . 8 ⊢ (𝐹 ⊆ 𝐻 ↔ (𝐹 ∪ 𝐻) = 𝐻)
26	24, 25	sylibr 224	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ 𝐻)
27		mreexexlem2d.5	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻))
28	27	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝑋 ∖ 𝐻))
29	26, 28	ssind 3837	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝐻 ∩ (𝑋 ∖ 𝐻)))
30		disjdif 4040	. . . . . 6 ⊢ (𝐻 ∩ (𝑋 ∖ 𝐻)) = ∅
31	29, 30	syl6sseq 3651	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ ∅)
32		ss0b 3973	. . . . 5 ⊢ (𝐹 ⊆ ∅ ↔ 𝐹 = ∅)
33	31, 32	sylib 208	. . . 4 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 = ∅)
34		mreexexlem3d.9	. . . 4 ⊢ (𝜑 → (𝐹 = ∅ ∨ 𝐺 = ∅))
35	1, 33, 34	mpjaodan 827	. . 3 ⊢ (𝜑 → 𝐹 = ∅)
36		0elpw 4834	. . 3 ⊢ ∅ ∈ 𝒫 𝐺
37	35, 36	syl6eqel 2709	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝒫 𝐺)
38	2	elfvexd 6222	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
39	27	difss2d 3740	. . . 4 ⊢ (𝜑 → 𝐹 ⊆ 𝑋)
40	38, 39	ssexd 4805	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
41		enrefg 7987	. . 3 ⊢ (𝐹 ∈ V → 𝐹 ≈ 𝐹)
42	40, 41	syl 17	. 2 ⊢ (𝜑 → 𝐹 ≈ 𝐹)
43		breq2 4657	. . . 4 ⊢ (𝑖 = 𝐹 → (𝐹 ≈ 𝑖 ↔ 𝐹 ≈ 𝐹))
44		uneq1 3760	. . . . 5 ⊢ (𝑖 = 𝐹 → (𝑖 ∪ 𝐻) = (𝐹 ∪ 𝐻))
45	44	eleq1d 2686	. . . 4 ⊢ (𝑖 = 𝐹 → ((𝑖 ∪ 𝐻) ∈ 𝐼 ↔ (𝐹 ∪ 𝐻) ∈ 𝐼))
46	43, 45	anbi12d 747	. . 3 ⊢ (𝑖 = 𝐹 → ((𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼) ↔ (𝐹 ≈ 𝐹 ∧ (𝐹 ∪ 𝐻) ∈ 𝐼)))
47	46	rspcev 3309	. 2 ⊢ ((𝐹 ∈ 𝒫 𝐺 ∧ (𝐹 ≈ 𝐹 ∧ (𝐹 ∪ 𝐻) ∈ 𝐼)) → ∃𝑖 ∈ 𝒫 𝐺(𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼))
48	37, 42, 16, 47	syl12anc 1324	1 ⊢ (𝜑 → ∃𝑖 ∈ 𝒫 𝐺(𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼))

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177   class class class wbr 4653  ‘cfv 5888   ≈ cen 7952  Moorecmre 16242  mrClscmrc 16243  mrIndcmri 16244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-en 7956  df-mre 16246  df-mrc 16247  df-mri 16248

This theorem is referenced by:  mreexexlem4d  16307  mreexexd  16308  mreexexdOLD  16309