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Theorem ismrcd2 41427
Description: Second half of ismrcd1 41426. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
ismrcd.b (πœ‘ β†’ 𝐡 ∈ 𝑉)
ismrcd.f (πœ‘ β†’ 𝐹:𝒫 π΅βŸΆπ’« 𝐡)
ismrcd.e ((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ π‘₯ βŠ† (πΉβ€˜π‘₯))
ismrcd.m ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ (πΉβ€˜π‘¦) βŠ† (πΉβ€˜π‘₯))
ismrcd.i ((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ (πΉβ€˜(πΉβ€˜π‘₯)) = (πΉβ€˜π‘₯))
Assertion
Ref Expression
ismrcd2 (πœ‘ β†’ 𝐹 = (mrClsβ€˜dom (𝐹 ∩ I )))
Distinct variable groups:   πœ‘,π‘₯,𝑦   π‘₯,𝐡,𝑦   π‘₯,𝐹,𝑦   π‘₯,𝑉,𝑦
Proof of Theorem ismrcd2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ismrcd.f . . 3 (πœ‘ β†’ 𝐹:𝒫 π΅βŸΆπ’« 𝐡)
21ffnd 6718 . 2 (πœ‘ β†’ 𝐹 Fn 𝒫 𝐡)
3 ismrcd.b . . . 4 (πœ‘ β†’ 𝐡 ∈ 𝑉)
4 ismrcd.e . . . 4 ((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ π‘₯ βŠ† (πΉβ€˜π‘₯))
5 ismrcd.m . . . 4 ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ (πΉβ€˜π‘¦) βŠ† (πΉβ€˜π‘₯))
6 ismrcd.i . . . 4 ((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ (πΉβ€˜(πΉβ€˜π‘₯)) = (πΉβ€˜π‘₯))
73, 1, 4, 5, 6ismrcd1 41426 . . 3 (πœ‘ β†’ dom (𝐹 ∩ I ) ∈ (Mooreβ€˜π΅))
8 eqid 2732 . . . 4 (mrClsβ€˜dom (𝐹 ∩ I )) = (mrClsβ€˜dom (𝐹 ∩ I ))
98mrcf 17552 . . 3 (dom (𝐹 ∩ I ) ∈ (Mooreβ€˜π΅) β†’ (mrClsβ€˜dom (𝐹 ∩ I )):𝒫 𝐡⟢dom (𝐹 ∩ I ))
10 ffn 6717 . . 3 ((mrClsβ€˜dom (𝐹 ∩ I )):𝒫 𝐡⟢dom (𝐹 ∩ I ) β†’ (mrClsβ€˜dom (𝐹 ∩ I )) Fn 𝒫 𝐡)
117, 9, 103syl 18 . 2 (πœ‘ β†’ (mrClsβ€˜dom (𝐹 ∩ I )) Fn 𝒫 𝐡)
127, 8mrcssvd 17566 . . . . . 6 (πœ‘ β†’ ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) βŠ† 𝐡)
1312adantr 481 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) βŠ† 𝐡)
14 elpwi 4609 . . . . . 6 (𝑧 ∈ 𝒫 𝐡 β†’ 𝑧 βŠ† 𝐡)
158mrcssid 17560 . . . . . 6 ((dom (𝐹 ∩ I ) ∈ (Mooreβ€˜π΅) ∧ 𝑧 βŠ† 𝐡) β†’ 𝑧 βŠ† ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§))
167, 14, 15syl2an 596 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ 𝑧 βŠ† ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§))
1753expib 1122 . . . . . . . 8 (πœ‘ β†’ ((π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ (πΉβ€˜π‘¦) βŠ† (πΉβ€˜π‘₯)))
1817alrimivv 1931 . . . . . . 7 (πœ‘ β†’ βˆ€π‘¦βˆ€π‘₯((π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ (πΉβ€˜π‘¦) βŠ† (πΉβ€˜π‘₯)))
19 vex 3478 . . . . . . . 8 𝑧 ∈ V
20 fvex 6904 . . . . . . . 8 ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) ∈ V
21 sseq1 4007 . . . . . . . . . . . 12 (π‘₯ = ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) β†’ (π‘₯ βŠ† 𝐡 ↔ ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) βŠ† 𝐡))
2221adantl 482 . . . . . . . . . . 11 ((𝑦 = 𝑧 ∧ π‘₯ = ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)) β†’ (π‘₯ βŠ† 𝐡 ↔ ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) βŠ† 𝐡))
23 sseq12 4009 . . . . . . . . . . 11 ((𝑦 = 𝑧 ∧ π‘₯ = ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)) β†’ (𝑦 βŠ† π‘₯ ↔ 𝑧 βŠ† ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)))
2422, 23anbi12d 631 . . . . . . . . . 10 ((𝑦 = 𝑧 ∧ π‘₯ = ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)) β†’ ((π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) ↔ (((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) βŠ† 𝐡 ∧ 𝑧 βŠ† ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§))))
25 fveq2 6891 . . . . . . . . . . 11 (𝑦 = 𝑧 β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘§))
26 fveq2 6891 . . . . . . . . . . 11 (π‘₯ = ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)))
27 sseq12 4009 . . . . . . . . . . 11 (((πΉβ€˜π‘¦) = (πΉβ€˜π‘§) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§))) β†’ ((πΉβ€˜π‘¦) βŠ† (πΉβ€˜π‘₯) ↔ (πΉβ€˜π‘§) βŠ† (πΉβ€˜((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§))))
2825, 26, 27syl2an 596 . . . . . . . . . 10 ((𝑦 = 𝑧 ∧ π‘₯ = ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)) β†’ ((πΉβ€˜π‘¦) βŠ† (πΉβ€˜π‘₯) ↔ (πΉβ€˜π‘§) βŠ† (πΉβ€˜((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§))))
2924, 28imbi12d 344 . . . . . . . . 9 ((𝑦 = 𝑧 ∧ π‘₯ = ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)) β†’ (((π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ (πΉβ€˜π‘¦) βŠ† (πΉβ€˜π‘₯)) ↔ ((((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) βŠ† 𝐡 ∧ 𝑧 βŠ† ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)) β†’ (πΉβ€˜π‘§) βŠ† (πΉβ€˜((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)))))
3029spc2gv 3590 . . . . . . . 8 ((𝑧 ∈ V ∧ ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) ∈ V) β†’ (βˆ€π‘¦βˆ€π‘₯((π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ (πΉβ€˜π‘¦) βŠ† (πΉβ€˜π‘₯)) β†’ ((((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) βŠ† 𝐡 ∧ 𝑧 βŠ† ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)) β†’ (πΉβ€˜π‘§) βŠ† (πΉβ€˜((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)))))
3119, 20, 30mp2an 690 . . . . . . 7 (βˆ€π‘¦βˆ€π‘₯((π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ (πΉβ€˜π‘¦) βŠ† (πΉβ€˜π‘₯)) β†’ ((((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) βŠ† 𝐡 ∧ 𝑧 βŠ† ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)) β†’ (πΉβ€˜π‘§) βŠ† (πΉβ€˜((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§))))
3218, 31syl 17 . . . . . 6 (πœ‘ β†’ ((((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) βŠ† 𝐡 ∧ 𝑧 βŠ† ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)) β†’ (πΉβ€˜π‘§) βŠ† (πΉβ€˜((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§))))
3332adantr 481 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ ((((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) βŠ† 𝐡 ∧ 𝑧 βŠ† ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)) β†’ (πΉβ€˜π‘§) βŠ† (πΉβ€˜((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§))))
3413, 16, 33mp2and 697 . . . 4 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ (πΉβ€˜π‘§) βŠ† (πΉβ€˜((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)))
358mrccl 17554 . . . . . 6 ((dom (𝐹 ∩ I ) ∈ (Mooreβ€˜π΅) ∧ 𝑧 βŠ† 𝐡) β†’ ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) ∈ dom (𝐹 ∩ I ))
367, 14, 35syl2an 596 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) ∈ dom (𝐹 ∩ I ))
372adantr 481 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ 𝐹 Fn 𝒫 𝐡)
3820elpw 4606 . . . . . . . 8 (((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) ∈ 𝒫 𝐡 ↔ ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) βŠ† 𝐡)
3912, 38sylibr 233 . . . . . . 7 (πœ‘ β†’ ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) ∈ 𝒫 𝐡)
4039adantr 481 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) ∈ 𝒫 𝐡)
41 fnelfp 7172 . . . . . 6 ((𝐹 Fn 𝒫 𝐡 ∧ ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) ∈ 𝒫 𝐡) β†’ (((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) ∈ dom (𝐹 ∩ I ) ↔ (πΉβ€˜((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)) = ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)))
4237, 40, 41syl2anc 584 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ (((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) ∈ dom (𝐹 ∩ I ) ↔ (πΉβ€˜((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)) = ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)))
4336, 42mpbid 231 . . . 4 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ (πΉβ€˜((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§)) = ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§))
4434, 43sseqtrd 4022 . . 3 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ (πΉβ€˜π‘§) βŠ† ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§))
457adantr 481 . . . 4 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ dom (𝐹 ∩ I ) ∈ (Mooreβ€˜π΅))
46 sseq1 4007 . . . . . . . 8 (π‘₯ = 𝑧 β†’ (π‘₯ βŠ† 𝐡 ↔ 𝑧 βŠ† 𝐡))
4746anbi2d 629 . . . . . . 7 (π‘₯ = 𝑧 β†’ ((πœ‘ ∧ π‘₯ βŠ† 𝐡) ↔ (πœ‘ ∧ 𝑧 βŠ† 𝐡)))
48 id 22 . . . . . . . 8 (π‘₯ = 𝑧 β†’ π‘₯ = 𝑧)
49 fveq2 6891 . . . . . . . 8 (π‘₯ = 𝑧 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘§))
5048, 49sseq12d 4015 . . . . . . 7 (π‘₯ = 𝑧 β†’ (π‘₯ βŠ† (πΉβ€˜π‘₯) ↔ 𝑧 βŠ† (πΉβ€˜π‘§)))
5147, 50imbi12d 344 . . . . . 6 (π‘₯ = 𝑧 β†’ (((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ π‘₯ βŠ† (πΉβ€˜π‘₯)) ↔ ((πœ‘ ∧ 𝑧 βŠ† 𝐡) β†’ 𝑧 βŠ† (πΉβ€˜π‘§))))
5251, 4chvarvv 2002 . . . . 5 ((πœ‘ ∧ 𝑧 βŠ† 𝐡) β†’ 𝑧 βŠ† (πΉβ€˜π‘§))
5314, 52sylan2 593 . . . 4 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ 𝑧 βŠ† (πΉβ€˜π‘§))
54 2fveq3 6896 . . . . . . . . 9 (π‘₯ = 𝑧 β†’ (πΉβ€˜(πΉβ€˜π‘₯)) = (πΉβ€˜(πΉβ€˜π‘§)))
5554, 49eqeq12d 2748 . . . . . . . 8 (π‘₯ = 𝑧 β†’ ((πΉβ€˜(πΉβ€˜π‘₯)) = (πΉβ€˜π‘₯) ↔ (πΉβ€˜(πΉβ€˜π‘§)) = (πΉβ€˜π‘§)))
5647, 55imbi12d 344 . . . . . . 7 (π‘₯ = 𝑧 β†’ (((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ (πΉβ€˜(πΉβ€˜π‘₯)) = (πΉβ€˜π‘₯)) ↔ ((πœ‘ ∧ 𝑧 βŠ† 𝐡) β†’ (πΉβ€˜(πΉβ€˜π‘§)) = (πΉβ€˜π‘§))))
5756, 6chvarvv 2002 . . . . . 6 ((πœ‘ ∧ 𝑧 βŠ† 𝐡) β†’ (πΉβ€˜(πΉβ€˜π‘§)) = (πΉβ€˜π‘§))
5814, 57sylan2 593 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ (πΉβ€˜(πΉβ€˜π‘§)) = (πΉβ€˜π‘§))
591ffvelcdmda 7086 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ (πΉβ€˜π‘§) ∈ 𝒫 𝐡)
60 fnelfp 7172 . . . . . 6 ((𝐹 Fn 𝒫 𝐡 ∧ (πΉβ€˜π‘§) ∈ 𝒫 𝐡) β†’ ((πΉβ€˜π‘§) ∈ dom (𝐹 ∩ I ) ↔ (πΉβ€˜(πΉβ€˜π‘§)) = (πΉβ€˜π‘§)))
6137, 59, 60syl2anc 584 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ ((πΉβ€˜π‘§) ∈ dom (𝐹 ∩ I ) ↔ (πΉβ€˜(πΉβ€˜π‘§)) = (πΉβ€˜π‘§)))
6258, 61mpbird 256 . . . 4 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ (πΉβ€˜π‘§) ∈ dom (𝐹 ∩ I ))
638mrcsscl 17563 . . . 4 ((dom (𝐹 ∩ I ) ∈ (Mooreβ€˜π΅) ∧ 𝑧 βŠ† (πΉβ€˜π‘§) ∧ (πΉβ€˜π‘§) ∈ dom (𝐹 ∩ I )) β†’ ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) βŠ† (πΉβ€˜π‘§))
6445, 53, 62, 63syl3anc 1371 . . 3 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§) βŠ† (πΉβ€˜π‘§))
6544, 64eqssd 3999 . 2 ((πœ‘ ∧ 𝑧 ∈ 𝒫 𝐡) β†’ (πΉβ€˜π‘§) = ((mrClsβ€˜dom (𝐹 ∩ I ))β€˜π‘§))
662, 11, 65eqfnfvd 7035 1 (πœ‘ β†’ 𝐹 = (mrClsβ€˜dom (𝐹 ∩ I )))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087  βˆ€wal 1539   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602   I cid 5573  dom cdm 5676   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  Moorecmre 17525  mrClscmrc 17526
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-mre 17529  df-mrc 17530
This theorem is referenced by:  istopclsd  41428  ismrc  41429
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