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Theorem ismrcd2 37788
Description: Second half of ismrcd1 37787. (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 6185 . . 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 37787 . . 3 (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
9 eqid 2771 . . . 4 (mrCls‘dom (𝐹 ∩ I )) = (mrCls‘dom (𝐹 ∩ I ))
109mrcf 16477 . . 3 (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )):𝒫 𝐵⟶dom (𝐹 ∩ I ))
11 ffn 6185 . . 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 16491 . . . . . 6 (𝜑 → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵)
1413adantr 466 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵)
15 elpwi 4307 . . . . . 6 (𝑧 ∈ 𝒫 𝐵𝑧𝐵)
169mrcssid 16485 . . . . . 6 ((dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) ∧ 𝑧𝐵) → 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
178, 15, 16syl2an 575 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → 𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
1863expib 1116 . . . . . . . 8 (𝜑 → ((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
1918alrimivv 2008 . . . . . . 7 (𝜑 → ∀𝑦𝑥((𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥)))
20 vex 3354 . . . . . . . 8 𝑧 ∈ V
21 fvex 6342 . . . . . . . 8 ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ V
22 sseq1 3775 . . . . . . . . . . . 12 (𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) → (𝑥𝐵 ↔ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵))
2322adantl 467 . . . . . . . . . . 11 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝑥𝐵 ↔ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵))
24 sseq12 3777 . . . . . . . . . . 11 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → (𝑦𝑥𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
2523, 24anbi12d 608 . . . . . . . . . 10 ((𝑦 = 𝑧𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) → ((𝑥𝐵𝑦𝑥) ↔ (((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵𝑧 ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
26 fveq2 6332 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
27 fveq2 6332 . . . . . . . . . . 11 (𝑥 = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) → (𝐹𝑥) = (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
28 sseq12 3777 . . . . . . . . . . 11 (((𝐹𝑦) = (𝐹𝑧) ∧ (𝐹𝑥) = (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))) → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧))))
2926, 27, 28syl2an 575 . . . . . . . . . 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 3447 . . . . . . . 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 664 . . . . . . 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 466 . . . . 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 671 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ⊆ (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)))
369mrccl 16479 . . . . . 6 ((dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) ∧ 𝑧𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ dom (𝐹 ∩ I ))
378, 15, 36syl2an 575 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ dom (𝐹 ∩ I ))
383adantr 466 . . . . . 6 ((𝜑𝑧 ∈ 𝒫 𝐵) → 𝐹 Fn 𝒫 𝐵)
3921elpw 4303 . . . . . . . 8 (((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵 ↔ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ 𝐵)
4013, 39sylibr 224 . . . . . . 7 (𝜑 → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵)
4140adantr 466 . . . . . 6 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ∈ 𝒫 𝐵)
42 fnelfp 6585 . . . . . 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 565 . . . . 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 222 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹‘((mrCls‘dom (𝐹 ∩ I ))‘𝑧)) = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
4535, 44sseqtrd 3790 . . 3 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ⊆ ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
468adantr 466 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
47 sseq1 3775 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
4847anbi2d 606 . . . . . . 7 (𝑥 = 𝑧 → ((𝜑𝑥𝐵) ↔ (𝜑𝑧𝐵)))
49 id 22 . . . . . . . 8 (𝑥 = 𝑧𝑥 = 𝑧)
50 fveq2 6332 . . . . . . . 8 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
5149, 50sseq12d 3783 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑧 ⊆ (𝐹𝑧)))
5248, 51imbi12d 333 . . . . . 6 (𝑥 = 𝑧 → (((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥)) ↔ ((𝜑𝑧𝐵) → 𝑧 ⊆ (𝐹𝑧))))
5352, 5chvarv 2425 . . . . 5 ((𝜑𝑧𝐵) → 𝑧 ⊆ (𝐹𝑧))
5415, 53sylan2 572 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → 𝑧 ⊆ (𝐹𝑧))
5550fveq2d 6336 . . . . . . . . 9 (𝑥 = 𝑧 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑧)))
5655, 50eqeq12d 2786 . . . . . . . 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 2425 . . . . . 6 ((𝜑𝑧𝐵) → (𝐹‘(𝐹𝑧)) = (𝐹𝑧))
5915, 58sylan2 572 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹‘(𝐹𝑧)) = (𝐹𝑧))
601ffvelrnda 6502 . . . . . 6 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ∈ 𝒫 𝐵)
61 fnelfp 6585 . . . . . 6 ((𝐹 Fn 𝒫 𝐵 ∧ (𝐹𝑧) ∈ 𝒫 𝐵) → ((𝐹𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
6238, 60, 61syl2anc 565 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((𝐹𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐹𝑧)) = (𝐹𝑧)))
6359, 62mpbird 247 . . . 4 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) ∈ dom (𝐹 ∩ I ))
649mrcsscl 16488 . . . 4 ((dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) ∧ 𝑧 ⊆ (𝐹𝑧) ∧ (𝐹𝑧) ∈ dom (𝐹 ∩ I )) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ (𝐹𝑧))
6546, 54, 63, 64syl3anc 1476 . . 3 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((mrCls‘dom (𝐹 ∩ I ))‘𝑧) ⊆ (𝐹𝑧))
6645, 65eqssd 3769 . 2 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐹𝑧) = ((mrCls‘dom (𝐹 ∩ I ))‘𝑧))
673, 12, 66eqfnfvd 6457 1 (𝜑𝐹 = (mrCls‘dom (𝐹 ∩ I )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071  wal 1629   = wceq 1631  wcel 2145  Vcvv 3351  cin 3722  wss 3723  𝒫 cpw 4297   I cid 5156  dom cdm 5249   Fn wfn 6026  wf 6027  cfv 6031  Moorecmre 16450  mrClscmrc 16451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-mre 16454  df-mrc 16455
This theorem is referenced by:  istopclsd  37789  ismrc  37790
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