Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ismrcd2 Structured version   Visualization version   GIF version

Theorem ismrcd2 38802
Description: Second half of ismrcd1 38801. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
ismrcd.b (𝜑𝐵𝑉)
ismrcd.f (𝜑𝐹:𝒫 𝐵⟶𝒫 𝐵)
ismrcd.e ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))
ismrcd.m ((𝜑𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥))
ismrcd.i ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))
Assertion
Ref Expression
ismrcd2 (𝜑𝐹 = (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.
StepHypRef Expression
1 ismrcd.f . . 3 (𝜑𝐹:𝒫 𝐵⟶𝒫 𝐵)
21ffnd 6390 . 2 (𝜑𝐹 Fn 𝒫 𝐵)
3 ismrcd.b . . . 4 (𝜑𝐵𝑉)
4 ismrcd.e . . . 4 ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))
5 ismrcd.m . . . 4 ((𝜑𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥))
6 ismrcd.i . . . 4 ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))
73, 1, 4, 5, 6ismrcd1 38801 . . 3 (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
8 eqid 2797 . . . 4 (mrCls‘dom (𝐹 ∩ I )) = (mrCls‘dom (𝐹 ∩ I ))
98mrcf 16713 . . 3 (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )):𝒫 𝐵⟶dom (𝐹 ∩ I ))
10 ffn 6389 . . 3 ((mrCls‘dom (𝐹 ∩ I )):𝒫 𝐵⟶dom (𝐹 ∩ I ) → (mrCls‘dom (𝐹 ∩ I )) Fn 𝒫 𝐵)
117, 9, 103syl 18 . 2 (𝜑 → (mrCls‘dom (𝐹 ∩ I )) Fn 𝒫 𝐵)
127, 8mrcssvd 16727 . . . . . 6 (𝜑 → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵)
1312adantr 481 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵)
14 elpwi 4469 . . . . . 6 (𝑧 ∈ 𝒫 𝐵𝑧𝐵)
158mrcssid 16721 . . . . . 6 ((dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) ∧ 𝑧𝐵) → 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
167, 14, 15syl2an 595 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
1753expib 1115 . . . . . . . 8 (𝜑 → ((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
1817alrimivv 1910 . . . . . . 7 (𝜑 → ∀𝑦𝑥((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
19 vex 3443 . . . . . . . 8 𝑧 ∈ V
20 fvex 6558 . . . . . . . 8 ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ V
21 sseq1 3919 . . . . . . . . . . . 12 (𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) → (𝑥𝐵 ↔ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵))
2221adantl 482 . . . . . . . . . . 11 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝑥𝐵 ↔ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵))
23 sseq12 3921 . . . . . . . . . . 11 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝑦𝑥𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
2422, 23anbi12d 630 . . . . . . . . . 10 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → ((𝑥𝐵𝑦𝑥) ↔ (((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
25 fveq2 6545 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
26 fveq2 6545 . . . . . . . . . . 11 (𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) → (𝐹𝑥) = (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
27 sseq12 3921 . . . . . . . . . . 11 (((𝐹𝑦) = (𝐹𝑧) ∧ (𝐹𝑥) = (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))) → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
2825, 26, 27syl2an 595 . . . . . . . . . 10 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
2924, 28imbi12d 346 . . . . . . . . 9 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)) ↔ ((((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))))
3029spc2gv 3545 . . . . . . . 8 ((𝑧 ∈ V ∧ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ V) → (∀𝑦𝑥((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)) → ((((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))))
3119, 20, 30mp2an 688 . . . . . . 7 (∀𝑦𝑥((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)) → ((((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
3218, 31syl 17 . . . . . 6 (𝜑 → ((((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
3332adantr 481 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
3413, 16, 33mp2and 695 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
358mrccl 16715 . . . . . 6 ((dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) ∧ 𝑧𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ dom (𝐹 ∩ I ))
367, 14, 35syl2an 595 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ dom (𝐹 ∩ I ))
372adantr 481 . . . . . 6 ((𝜑𝑧 ∈ 𝒫 𝐵) → 𝐹 Fn 𝒫 𝐵)
3820elpw 4465 . . . . . . . 8 (((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵 ↔ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵)
3912, 38sylibr 235 . . . . . . 7 (𝜑 → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵)
4039adantr 481 . . . . . 6 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵)
41 fnelfp 6807 . . . . . 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)))
4237, 40, 41syl2anc 584 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → (((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
4336, 42mpbid 233 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
4434, 43sseqtrd 3934 . . 3 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
457adantr 481 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
46 sseq1 3919 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
4746anbi2d 628 . . . . . . 7 (𝑥 = 𝑧 → ((𝜑𝑥𝐵) ↔ (𝜑𝑧𝐵)))
48 id 22 . . . . . . . 8 (𝑥 = 𝑧𝑥 = 𝑧)
49 fveq2 6545 . . . . . . . 8 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
5048, 49sseq12d 3927 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑧 ⊆ (𝐹𝑧)))
5147, 50imbi12d 346 . . . . . 6 (𝑥 = 𝑧 → (((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥)) ↔ ((𝜑𝑧𝐵) → 𝑧 ⊆ (𝐹𝑧))))
5251, 4chvarv 2372 . . . . 5 ((𝜑𝑧𝐵) → 𝑧 ⊆ (𝐹𝑧))
5314, 52sylan2 592 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → 𝑧 ⊆ (𝐹𝑧))
54 2fveq3 6550 . . . . . . . . 9 (𝑥 = 𝑧 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑧)))
5554, 49eqeq12d 2812 . . . . . . . 8 (𝑥 = 𝑧 → ((𝐹‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
5647, 55imbi12d 346 . . . . . . 7 (𝑥 = 𝑧 → (((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥)) ↔ ((𝜑𝑧𝐵) → (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
5756, 6chvarv 2372 . . . . . 6 ((𝜑𝑧𝐵) → (𝐹‘(𝐹𝑧)) = (𝐹𝑧))
5814, 57sylan2 592 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹‘(𝐹𝑧)) = (𝐹𝑧))
591ffvelrnda 6723 . . . . . 6 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ∈ 𝒫 𝐵)
60 fnelfp 6807 . . . . . 6 ((𝐹 Fn 𝒫 𝐵 ∧ (𝐹𝑧) ∈ 𝒫 𝐵) → ((𝐹𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
6137, 59, 60syl2anc 584 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((𝐹𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
6258, 61mpbird 258 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ∈ dom (𝐹 ∩ I ))
638mrcsscl 16724 . . . 4 ((dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) ∧ 𝑧 ⊆ (𝐹𝑧) ∧ (𝐹𝑧) ∈ dom (𝐹 ∩ I )) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ (𝐹𝑧))
6445, 53, 62, 63syl3anc 1364 . . 3 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ (𝐹𝑧))
6544, 64eqssd 3912 . 2 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
662, 11, 65eqfnfvd 6677 1 (𝜑𝐹 = (mrCls‘dom (𝐹 ∩ I )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080  wal 1523   = wceq 1525  wcel 2083  Vcvv 3440  cin 3864  wss 3865  𝒫 cpw 4459   I cid 5354  dom cdm 5450   Fn wfn 6227  wf 6228  cfv 6232  Moorecmre 16686  mrClscmrc 16687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-int 4789  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-fv 6240  df-mre 16690  df-mrc 16691
This theorem is referenced by:  istopclsd  38803  ismrc  38804
  Copyright terms: Public domain W3C validator