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Theorem ismrcd2 39627
Description: Second half of ismrcd1 39626. (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 6492 . 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 39626 . . 3 (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
8 eqid 2801 . . . 4 (mrCls‘dom (𝐹 ∩ I )) = (mrCls‘dom (𝐹 ∩ I ))
98mrcf 16875 . . 3 (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )):𝒫 𝐵⟶dom (𝐹 ∩ I ))
10 ffn 6491 . . 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 16889 . . . . . 6 (𝜑 → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵)
1312adantr 484 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵)
14 elpwi 4509 . . . . . 6 (𝑧 ∈ 𝒫 𝐵𝑧𝐵)
158mrcssid 16883 . . . . . 6 ((dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) ∧ 𝑧𝐵) → 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
167, 14, 15syl2an 598 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
1753expib 1119 . . . . . . . 8 (𝜑 → ((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
1817alrimivv 1929 . . . . . . 7 (𝜑 → ∀𝑦𝑥((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
19 vex 3447 . . . . . . . 8 𝑧 ∈ V
20 fvex 6662 . . . . . . . 8 ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ V
21 sseq1 3943 . . . . . . . . . . . 12 (𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) → (𝑥𝐵 ↔ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵))
2221adantl 485 . . . . . . . . . . 11 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝑥𝐵 ↔ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵))
23 sseq12 3945 . . . . . . . . . . 11 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝑦𝑥𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
2422, 23anbi12d 633 . . . . . . . . . 10 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → ((𝑥𝐵𝑦𝑥) ↔ (((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
25 fveq2 6649 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
26 fveq2 6649 . . . . . . . . . . 11 (𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) → (𝐹𝑥) = (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
27 sseq12 3945 . . . . . . . . . . 11 (((𝐹𝑦) = (𝐹𝑧) ∧ (𝐹𝑥) = (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))) → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
2825, 26, 27syl2an 598 . . . . . . . . . 10 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
2924, 28imbi12d 348 . . . . . . . . 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 3552 . . . . . . . 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 691 . . . . . . 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 484 . . . . 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 698 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
358mrccl 16877 . . . . . 6 ((dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) ∧ 𝑧𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ dom (𝐹 ∩ I ))
367, 14, 35syl2an 598 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ dom (𝐹 ∩ I ))
372adantr 484 . . . . . 6 ((𝜑𝑧 ∈ 𝒫 𝐵) → 𝐹 Fn 𝒫 𝐵)
3820elpw 4504 . . . . . . . 8 (((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵 ↔ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵)
3912, 38sylibr 237 . . . . . . 7 (𝜑 → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵)
4039adantr 484 . . . . . 6 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵)
41 fnelfp 6918 . . . . . 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 587 . . . . 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 235 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
4434, 43sseqtrd 3958 . . 3 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
457adantr 484 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
46 sseq1 3943 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
4746anbi2d 631 . . . . . . 7 (𝑥 = 𝑧 → ((𝜑𝑥𝐵) ↔ (𝜑𝑧𝐵)))
48 id 22 . . . . . . . 8 (𝑥 = 𝑧𝑥 = 𝑧)
49 fveq2 6649 . . . . . . . 8 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
5048, 49sseq12d 3951 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑧 ⊆ (𝐹𝑧)))
5147, 50imbi12d 348 . . . . . 6 (𝑥 = 𝑧 → (((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥)) ↔ ((𝜑𝑧𝐵) → 𝑧 ⊆ (𝐹𝑧))))
5251, 4chvarvv 2005 . . . . 5 ((𝜑𝑧𝐵) → 𝑧 ⊆ (𝐹𝑧))
5314, 52sylan2 595 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → 𝑧 ⊆ (𝐹𝑧))
54 2fveq3 6654 . . . . . . . . 9 (𝑥 = 𝑧 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑧)))
5554, 49eqeq12d 2817 . . . . . . . 8 (𝑥 = 𝑧 → ((𝐹‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
5647, 55imbi12d 348 . . . . . . 7 (𝑥 = 𝑧 → (((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥)) ↔ ((𝜑𝑧𝐵) → (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
5756, 6chvarvv 2005 . . . . . 6 ((𝜑𝑧𝐵) → (𝐹‘(𝐹𝑧)) = (𝐹𝑧))
5814, 57sylan2 595 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹‘(𝐹𝑧)) = (𝐹𝑧))
591ffvelrnda 6832 . . . . . 6 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ∈ 𝒫 𝐵)
60 fnelfp 6918 . . . . . 6 ((𝐹 Fn 𝒫 𝐵 ∧ (𝐹𝑧) ∈ 𝒫 𝐵) → ((𝐹𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
6137, 59, 60syl2anc 587 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((𝐹𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
6258, 61mpbird 260 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ∈ dom (𝐹 ∩ I ))
638mrcsscl 16886 . . . 4 ((dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) ∧ 𝑧 ⊆ (𝐹𝑧) ∧ (𝐹𝑧) ∈ dom (𝐹 ∩ I )) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ (𝐹𝑧))
6445, 53, 62, 63syl3anc 1368 . . 3 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ (𝐹𝑧))
6544, 64eqssd 3935 . 2 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
662, 11, 65eqfnfvd 6786 1 (𝜑𝐹 = (mrCls‘dom (𝐹 ∩ I )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wal 1536   = wceq 1538  wcel 2112  Vcvv 3444  cin 3883  wss 3884  𝒫 cpw 4500   I cid 5427  dom cdm 5523   Fn wfn 6323  wf 6324  cfv 6328  Moorecmre 16848  mrClscmrc 16849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-mre 16852  df-mrc 16853
This theorem is referenced by:  istopclsd  39628  ismrc  39629
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