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Theorem ismrcd2 36280
Description: Second half of ismrcd1 36279. (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)
2 ffn 5958 . . 3 (𝐹:𝒫 𝐵⟶𝒫 𝐵𝐹 Fn 𝒫 𝐵)
31, 2syl 17 . 2 (𝜑𝐹 Fn 𝒫 𝐵)
4 ismrcd.b . . . 4 (𝜑𝐵𝑉)
5 ismrcd.e . . . 4 ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))
6 ismrcd.m . . . 4 ((𝜑𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥))
7 ismrcd.i . . . 4 ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))
84, 1, 5, 6, 7ismrcd1 36279 . . 3 (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
9 eqid 2610 . . . 4 (mrCls‘dom (𝐹 ∩ I )) = (mrCls‘dom (𝐹 ∩ I ))
109mrcf 16092 . . 3 (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )):𝒫 𝐵⟶dom (𝐹 ∩ I ))
11 ffn 5958 . . 3 ((mrCls‘dom (𝐹 ∩ I )):𝒫 𝐵⟶dom (𝐹 ∩ I ) → (mrCls‘dom (𝐹 ∩ I )) Fn 𝒫 𝐵)
128, 10, 113syl 18 . 2 (𝜑 → (mrCls‘dom (𝐹 ∩ I )) Fn 𝒫 𝐵)
138, 9mrcssvd 16106 . . . . . 6 (𝜑 → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵)
1413adantr 480 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵)
15 elpwi 4117 . . . . . 6 (𝑧 ∈ 𝒫 𝐵𝑧𝐵)
169mrcssid 16100 . . . . . 6 ((dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) ∧ 𝑧𝐵) → 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
178, 15, 16syl2an 493 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
1863expib 1260 . . . . . . . 8 (𝜑 → ((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
1918alrimivv 1843 . . . . . . 7 (𝜑 → ∀𝑦𝑥((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
20 vex 3176 . . . . . . . 8 𝑧 ∈ V
21 fvex 6113 . . . . . . . 8 ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ V
22 sseq1 3589 . . . . . . . . . . . 12 (𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) → (𝑥𝐵 ↔ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵))
2322adantl 481 . . . . . . . . . . 11 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝑥𝐵 ↔ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵))
24 sseq12 3591 . . . . . . . . . . 11 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝑦𝑥𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
2523, 24anbi12d 743 . . . . . . . . . 10 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → ((𝑥𝐵𝑦𝑥) ↔ (((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
26 fveq2 6103 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
27 fveq2 6103 . . . . . . . . . . 11 (𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) → (𝐹𝑥) = (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
28 sseq12 3591 . . . . . . . . . . 11 (((𝐹𝑦) = (𝐹𝑧) ∧ (𝐹𝑥) = (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))) → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
2926, 27, 28syl2an 493 . . . . . . . . . 10 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
3025, 29imbi12d 333 . . . . . . . . 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)))))
3130spc2gv 3269 . . . . . . . 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)))))
3220, 21, 31mp2an 704 . . . . . . 7 (∀𝑦𝑥((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)) → ((((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
3319, 32syl 17 . . . . . 6 (𝜑 → ((((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
3433adantr 480 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
3514, 17, 34mp2and 711 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
369mrccl 16094 . . . . . 6 ((dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) ∧ 𝑧𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ dom (𝐹 ∩ I ))
378, 15, 36syl2an 493 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ dom (𝐹 ∩ I ))
383adantr 480 . . . . . 6 ((𝜑𝑧 ∈ 𝒫 𝐵) → 𝐹 Fn 𝒫 𝐵)
3921elpw 4114 . . . . . . . 8 (((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵 ↔ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵)
4013, 39sylibr 223 . . . . . . 7 (𝜑 → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵)
4140adantr 480 . . . . . 6 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵)
42 fnelfp 6346 . . . . . 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)))
4338, 41, 42syl2anc 691 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → (((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
4437, 43mpbid 221 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
4535, 44sseqtrd 3604 . . 3 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
468adantr 480 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
47 sseq1 3589 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
4847anbi2d 736 . . . . . . 7 (𝑥 = 𝑧 → ((𝜑𝑥𝐵) ↔ (𝜑𝑧𝐵)))
49 id 22 . . . . . . . 8 (𝑥 = 𝑧𝑥 = 𝑧)
50 fveq2 6103 . . . . . . . 8 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
5149, 50sseq12d 3597 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑧 ⊆ (𝐹𝑧)))
5248, 51imbi12d 333 . . . . . 6 (𝑥 = 𝑧 → (((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥)) ↔ ((𝜑𝑧𝐵) → 𝑧 ⊆ (𝐹𝑧))))
5352, 5chvarv 2251 . . . . 5 ((𝜑𝑧𝐵) → 𝑧 ⊆ (𝐹𝑧))
5415, 53sylan2 490 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → 𝑧 ⊆ (𝐹𝑧))
5550fveq2d 6107 . . . . . . . . 9 (𝑥 = 𝑧 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑧)))
5655, 50eqeq12d 2625 . . . . . . . 8 (𝑥 = 𝑧 → ((𝐹‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
5748, 56imbi12d 333 . . . . . . 7 (𝑥 = 𝑧 → (((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥)) ↔ ((𝜑𝑧𝐵) → (𝐹‘(𝐹𝑧)) = (𝐹𝑧))))
5857, 7chvarv 2251 . . . . . 6 ((𝜑𝑧𝐵) → (𝐹‘(𝐹𝑧)) = (𝐹𝑧))
5915, 58sylan2 490 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹‘(𝐹𝑧)) = (𝐹𝑧))
601ffvelrnda 6267 . . . . . 6 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ∈ 𝒫 𝐵)
61 fnelfp 6346 . . . . . 6 ((𝐹 Fn 𝒫 𝐵 ∧ (𝐹𝑧) ∈ 𝒫 𝐵) → ((𝐹𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
6238, 60, 61syl2anc 691 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((𝐹𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
6359, 62mpbird 246 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ∈ dom (𝐹 ∩ I ))
649mrcsscl 16103 . . . 4 ((dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) ∧ 𝑧 ⊆ (𝐹𝑧) ∧ (𝐹𝑧) ∈ dom (𝐹 ∩ I )) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ (𝐹𝑧))
6546, 54, 63, 64syl3anc 1318 . . 3 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ (𝐹𝑧))
6645, 65eqssd 3585 . 2 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
673, 12, 66eqfnfvd 6222 1 (𝜑𝐹 = (mrCls‘dom (𝐹 ∩ I )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031  wal 1473   = wceq 1475  wcel 1977  Vcvv 3173  cin 3539  wss 3540  𝒫 cpw 4108   I cid 4948  dom cdm 5038   Fn wfn 5799  wf 5800  cfv 5804  Moorecmre 16065  mrClscmrc 16066
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-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-fv 5812  df-mre 16069  df-mrc 16070
This theorem is referenced by:  istopclsd  36281  ismrc  36282
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