Theorem ismrcd2 40165

Description: Second half of ismrcd1 40164. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Hypotheses

Ref	Expression
ismrcd.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
ismrcd.f	⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵)
ismrcd.e	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
ismrcd.m	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥))
ismrcd.i	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))

Assertion

Ref	Expression
ismrcd2	⊢ (𝜑 → 𝐹 = (mrCls‘dom (𝐹 ∩ I )))

Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝑉,𝑦

Proof of Theorem ismrcd2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismrcd.f	. . 3 ⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵)
2	1	ffnd 6524	. 2 ⊢ (𝜑 → 𝐹 Fn 𝒫 𝐵)
3		ismrcd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
4		ismrcd.e	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
5		ismrcd.m	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥))
6		ismrcd.i	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))
7	3, 1, 4, 5, 6	ismrcd1 40164	. . 3 ⊢ (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
8		eqid 2736	. . . 4 ⊢ (mrCls‘dom (𝐹 ∩ I )) = (mrCls‘dom (𝐹 ∩ I ))
9	8	mrcf 17066	. . 3 ⊢ (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )):𝒫 𝐵⟶dom (𝐹 ∩ I ))
10		ffn 6523	. . 3 ⊢ ((mrCls‘dom (𝐹 ∩ I )):𝒫 𝐵⟶dom (𝐹 ∩ I ) → (mrCls‘dom (𝐹 ∩ I )) Fn 𝒫 𝐵)
11	7, 9, 10	3syl 18	. 2 ⊢ (𝜑 → (mrCls‘dom (𝐹 ∩ I )) Fn 𝒫 𝐵)
12	7, 8	mrcssvd 17080	. . . . . 6 ⊢ (𝜑 → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵)
13	12	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵)
14		elpwi 4508	. . . . . 6 ⊢ (𝑧 ∈ 𝒫 𝐵 → 𝑧 ⊆ 𝐵)
15	8	mrcssid 17074	. . . . . 6 ⊢ ((dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) ∧ 𝑧 ⊆ 𝐵) → 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
16	7, 14, 15	syl2an 599	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
17	5	3expib 1124	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
18	17	alrimivv 1936	. . . . . . 7 ⊢ (𝜑 → ∀𝑦∀𝑥((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
19		vex 3402	. . . . . . . 8 ⊢ 𝑧 ∈ V
20		fvex 6708	. . . . . . . 8 ⊢ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ V
21		sseq1 3912	. . . . . . . . . . . 12 ⊢ (𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) → (𝑥 ⊆ 𝐵 ↔ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵))
22	21	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝑥 ⊆ 𝐵 ↔ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵))
23		sseq12 3914	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝑦 ⊆ 𝑥 ↔ 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
24	22, 23	anbi12d 634	. . . . . . . . . 10 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) ↔ (((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵 ∧ 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
25		fveq2 6695	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
26		fveq2 6695	. . . . . . . . . . 11 ⊢ (𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) → (𝐹‘𝑥) = (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
27		sseq12 3914	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑦) = (𝐹‘𝑧) ∧ (𝐹‘𝑥) = (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))) → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
28	25, 26, 27	syl2an 599	. . . . . . . . . 10 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
29	24, 28	imbi12d 348	. . . . . . . . 9 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ↔ ((((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵 ∧ 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝐹‘𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))))
30	29	spc2gv 3505	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ V) → (∀𝑦∀𝑥((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ((((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵 ∧ 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝐹‘𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))))
31	19, 20, 30	mp2an 692	. . . . . . 7 ⊢ (∀𝑦∀𝑥((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ((((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵 ∧ 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝐹‘𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
32	18, 31	syl 17	. . . . . 6 ⊢ (𝜑 → ((((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵 ∧ 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝐹‘𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
33	32	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → ((((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵 ∧ 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝐹‘𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
34	13, 16, 33	mp2and 699	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → (𝐹‘𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
35	8	mrccl 17068	. . . . . 6 ⊢ ((dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) ∧ 𝑧 ⊆ 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ dom (𝐹 ∩ I ))
36	7, 14, 35	syl2an 599	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ dom (𝐹 ∩ I ))
37	2	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → 𝐹 Fn 𝒫 𝐵)
38	20	elpw 4503	. . . . . . . 8 ⊢ (((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵 ↔ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵)
39	12, 38	sylibr 237	. . . . . . 7 ⊢ (𝜑 → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵)
40	39	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵)
41		fnelfp 6968	. . . . . 6 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵) → (((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
42	37, 40, 41	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → (((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
43	36, 42	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
44	34, 43	sseqtrd 3927	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → (𝐹‘𝑧) ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
45	7	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
46		sseq1 3912	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝐵 ↔ 𝑧 ⊆ 𝐵))
47	46	anbi2d 632	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ 𝑥 ⊆ 𝐵) ↔ (𝜑 ∧ 𝑧 ⊆ 𝐵)))
48		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
49		fveq2 6695	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
50	48, 49	sseq12d 3920	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ (𝐹‘𝑥) ↔ 𝑧 ⊆ (𝐹‘𝑧)))
51	47, 50	imbi12d 348	. . . . . 6 ⊢ (𝑥 = 𝑧 → (((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → 𝑧 ⊆ (𝐹‘𝑧))))
52	51, 4	chvarvv 2008	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → 𝑧 ⊆ (𝐹‘𝑧))
53	14, 52	sylan2 596	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → 𝑧 ⊆ (𝐹‘𝑧))
54		2fveq3 6700	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑧)))
55	54, 49	eqeq12d 2752	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥) ↔ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧)))
56	47, 55	imbi12d 348	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧))))
57	56, 6	chvarvv 2008	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧))
58	14, 57	sylan2 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧))
59	1	ffvelrnda 6882	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → (𝐹‘𝑧) ∈ 𝒫 𝐵)
60		fnelfp 6968	. . . . . 6 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ (𝐹‘𝑧) ∈ 𝒫 𝐵) → ((𝐹‘𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧)))
61	37, 59, 60	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → ((𝐹‘𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧)))
62	58, 61	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → (𝐹‘𝑧) ∈ dom (𝐹 ∩ I ))
63	8	mrcsscl 17077	. . . 4 ⊢ ((dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) ∧ 𝑧 ⊆ (𝐹‘𝑧) ∧ (𝐹‘𝑧) ∈ dom (𝐹 ∩ I )) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ (𝐹‘𝑧))
64	45, 53, 62, 63	syl3anc 1373	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ (𝐹‘𝑧))
65	44, 64	eqssd 3904	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → (𝐹‘𝑧) = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
66	2, 11, 65	eqfnfvd 6833	1 ⊢ (𝜑 → 𝐹 = (mrCls‘dom (𝐹 ∩ I )))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089  ∀wal 1541   = wceq 1543   ∈ wcel 2112  Vcvv 3398   ∩ cin 3852   ⊆ wss 3853  𝒫 cpw 4499   I cid 5439  dom cdm 5536   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  Moorecmre 17039  mrClscmrc 17040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-int 4846  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-mre 17043  df-mrc 17044

This theorem is referenced by:  istopclsd  40166  ismrc  40167