Theorem mreexexlem3d 16129

Description: Base case of the induction in mreexexd 16131. (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 476	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = ∅) → 𝐹 = ∅)
2		mreexexlem2d.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
3	2	adantr 480	. . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
8		simpr 476	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐺 = ∅)
9	8	uneq1d 3728	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐺 ∪ 𝐻) = (∅ ∪ 𝐻))
10		uncom 3719	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∪ ∅) = (∅ ∪ 𝐻)
11		un0 3919	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∪ ∅) = 𝐻
12	10, 11	eqtr3i 2634	. . . . . . . . . . . . 13 ⊢ (∅ ∪ 𝐻) = 𝐻
13	9, 12	syl6eq 2660	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐺 ∪ 𝐻) = 𝐻)
14	13	fveq2d 6107	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝑁‘(𝐺 ∪ 𝐻)) = (𝑁‘𝐻))
15	7, 14	sseqtrd 3604	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝑁‘𝐻))
16		mreexexlem2d.8	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼)
17	16	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) ∈ 𝐼)
18	5, 3, 17	mrissd 16119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) ⊆ 𝑋)
19	18	unssbd 3753	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐻 ⊆ 𝑋)
20	3, 4, 19	mrcssidd 16108	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐻 ⊆ (𝑁‘𝐻))
21	15, 20	unssd 3751	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) ⊆ (𝑁‘𝐻))
22		ssun2 3739	. . . . . . . . . 10 ⊢ 𝐻 ⊆ (𝐹 ∪ 𝐻)
23	22	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐻 ⊆ (𝐹 ∪ 𝐻))
24	3, 4, 5, 21, 23, 17	mrissmrcd 16123	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) = 𝐻)
25		ssequn1 3745	. . . . . . . 8 ⊢ (𝐹 ⊆ 𝐻 ↔ (𝐹 ∪ 𝐻) = 𝐻)
26	24, 25	sylibr 223	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ 𝐻)
27		mreexexlem2d.5	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻))
28	27	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝑋 ∖ 𝐻))
29	26, 28	ssind 3799	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝐻 ∩ (𝑋 ∖ 𝐻)))
30		disjdif 3992	. . . . . 6 ⊢ (𝐻 ∩ (𝑋 ∖ 𝐻)) = ∅
31	29, 30	syl6sseq 3614	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ ∅)
32		ss0b 3925	. . . . 5 ⊢ (𝐹 ⊆ ∅ ↔ 𝐹 = ∅)
33	31, 32	sylib 207	. . . 4 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 = ∅)
34		mreexexlem3d.9	. . . 4 ⊢ (𝜑 → (𝐹 = ∅ ∨ 𝐺 = ∅))
35	1, 33, 34	mpjaodan 823	. . 3 ⊢ (𝜑 → 𝐹 = ∅)
36		0elpw 4760	. . 3 ⊢ ∅ ∈ 𝒫 𝐺
37	35, 36	syl6eqel 2696	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝒫 𝐺)
38	2	elfvexd 6132	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
39	27	difss2d 3702	. . . 4 ⊢ (𝜑 → 𝐹 ⊆ 𝑋)
40	38, 39	ssexd 4733	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
41		enrefg 7873	. . 3 ⊢ (𝐹 ∈ V → 𝐹 ≈ 𝐹)
42	40, 41	syl 17	. 2 ⊢ (𝜑 → 𝐹 ≈ 𝐹)
43		breq2 4587	. . . 4 ⊢ (𝑖 = 𝐹 → (𝐹 ≈ 𝑖 ↔ 𝐹 ≈ 𝐹))
44		uneq1 3722	. . . . 5 ⊢ (𝑖 = 𝐹 → (𝑖 ∪ 𝐻) = (𝐹 ∪ 𝐻))
45	44	eleq1d 2672	. . . 4 ⊢ (𝑖 = 𝐹 → ((𝑖 ∪ 𝐻) ∈ 𝐼 ↔ (𝐹 ∪ 𝐻) ∈ 𝐼))
46	43, 45	anbi12d 743	. . 3 ⊢ (𝑖 = 𝐹 → ((𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼) ↔ (𝐹 ≈ 𝐹 ∧ (𝐹 ∪ 𝐻) ∈ 𝐼)))
47	46	rspcev 3282	. 2 ⊢ ((𝐹 ∈ 𝒫 𝐺 ∧ (𝐹 ≈ 𝐹 ∧ (𝐹 ∪ 𝐻) ∈ 𝐼)) → ∃𝑖 ∈ 𝒫 𝐺(𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼))
48	37, 42, 16, 47	syl12anc 1316	1 ⊢ (𝜑 → ∃𝑖 ∈ 𝒫 𝐺(𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼))

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  ‘cfv 5804   ≈ cen 7838  Moorecmre 16065  mrClscmrc 16066  mrIndcmri 16067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-en 7842  df-mre 16069  df-mrc 16070  df-mri 16071

This theorem is referenced by:  mreexexlem4d  16130  mreexexd  16131  mreexexdOLD  16132