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Theorem isf34lem7 10249
Description: Lemma for isfin3-4 10252. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypothesis
Ref Expression
compss.a 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
Assertion
Ref Expression
isf34lem7 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ran 𝐺 ∈ ran 𝐺)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐹   𝑦,𝐺
Allowed substitution hints:   𝐹(𝑥)   𝐺(𝑥)

Proof of Theorem isf34lem7
StepHypRef Expression
1 compss.a . . . . . . 7 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
21isf34lem2 10243 . . . . . 6 (𝐴 ∈ FinIII𝐹:𝒫 𝐴⟶𝒫 𝐴)
32adantr 482 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
433adant3 1133 . . . 4 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
54ffnd 6665 . . 3 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐹 Fn 𝒫 𝐴)
6 imassrn 6021 . . . 4 (𝐹 “ ran 𝐺) ⊆ ran 𝐹
73frnd 6672 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ran 𝐹 ⊆ 𝒫 𝐴)
873adant3 1133 . . . 4 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ran 𝐹 ⊆ 𝒫 𝐴)
96, 8sstrid 3954 . . 3 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴)
10 simp1 1137 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐴 ∈ FinIII)
11 fco 6688 . . . . . . 7 ((𝐹:𝒫 𝐴⟶𝒫 𝐴𝐺:ω⟶𝒫 𝐴) → (𝐹𝐺):ω⟶𝒫 𝐴)
122, 11sylan 581 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹𝐺):ω⟶𝒫 𝐴)
13123adant3 1133 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹𝐺):ω⟶𝒫 𝐴)
14 sscon 4097 . . . . . . . 8 ((𝐺𝑦) ⊆ (𝐺‘suc 𝑦) → (𝐴 ∖ (𝐺‘suc 𝑦)) ⊆ (𝐴 ∖ (𝐺𝑦)))
15 simpr 486 . . . . . . . . . . 11 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → 𝐺:ω⟶𝒫 𝐴)
16 peano2 7818 . . . . . . . . . . 11 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
17 fvco3 6936 . . . . . . . . . . 11 ((𝐺:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → ((𝐹𝐺)‘suc 𝑦) = (𝐹‘(𝐺‘suc 𝑦)))
1815, 16, 17syl2an 597 . . . . . . . . . 10 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹𝐺)‘suc 𝑦) = (𝐹‘(𝐺‘suc 𝑦)))
19 simpll 766 . . . . . . . . . . 11 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → 𝐴 ∈ FinIII)
20 ffvelcdm 7028 . . . . . . . . . . . . 13 ((𝐺:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ∈ 𝒫 𝐴)
2115, 16, 20syl2an 597 . . . . . . . . . . . 12 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ∈ 𝒫 𝐴)
2221elpwid 4568 . . . . . . . . . . 11 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ⊆ 𝐴)
231isf34lem1 10242 . . . . . . . . . . 11 ((𝐴 ∈ FinIII ∧ (𝐺‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝐺‘suc 𝑦)) = (𝐴 ∖ (𝐺‘suc 𝑦)))
2419, 22, 23syl2anc 585 . . . . . . . . . 10 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐹‘(𝐺‘suc 𝑦)) = (𝐴 ∖ (𝐺‘suc 𝑦)))
2518, 24eqtrd 2778 . . . . . . . . 9 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹𝐺)‘suc 𝑦) = (𝐴 ∖ (𝐺‘suc 𝑦)))
26 fvco3 6936 . . . . . . . . . . 11 ((𝐺:ω⟶𝒫 𝐴𝑦 ∈ ω) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
2726adantll 713 . . . . . . . . . 10 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
28 ffvelcdm 7028 . . . . . . . . . . . . 13 ((𝐺:ω⟶𝒫 𝐴𝑦 ∈ ω) → (𝐺𝑦) ∈ 𝒫 𝐴)
2928adantll 713 . . . . . . . . . . . 12 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺𝑦) ∈ 𝒫 𝐴)
3029elpwid 4568 . . . . . . . . . . 11 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺𝑦) ⊆ 𝐴)
311isf34lem1 10242 . . . . . . . . . . 11 ((𝐴 ∈ FinIII ∧ (𝐺𝑦) ⊆ 𝐴) → (𝐹‘(𝐺𝑦)) = (𝐴 ∖ (𝐺𝑦)))
3219, 30, 31syl2anc 585 . . . . . . . . . 10 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐹‘(𝐺𝑦)) = (𝐴 ∖ (𝐺𝑦)))
3327, 32eqtrd 2778 . . . . . . . . 9 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹𝐺)‘𝑦) = (𝐴 ∖ (𝐺𝑦)))
3425, 33sseq12d 3976 . . . . . . . 8 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦) ↔ (𝐴 ∖ (𝐺‘suc 𝑦)) ⊆ (𝐴 ∖ (𝐺𝑦))))
3514, 34syl5ibr 246 . . . . . . 7 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐺𝑦) ⊆ (𝐺‘suc 𝑦) → ((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦)))
3635ralimdva 3163 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦) → ∀𝑦 ∈ ω ((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦)))
37363impia 1118 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ∀𝑦 ∈ ω ((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦))
38 fin33i 10239 . . . . 5 ((𝐴 ∈ FinIII ∧ (𝐹𝐺):ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω ((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦)) → ran (𝐹𝐺) ∈ ran (𝐹𝐺))
3910, 13, 37, 38syl3anc 1372 . . . 4 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ran (𝐹𝐺) ∈ ran (𝐹𝐺))
40 rnco2 6202 . . . . 5 ran (𝐹𝐺) = (𝐹 “ ran 𝐺)
4140inteqi 4910 . . . 4 ran (𝐹𝐺) = (𝐹 “ ran 𝐺)
4239, 41, 403eltr3g 2855 . . 3 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 “ ran 𝐺) ∈ (𝐹 “ ran 𝐺))
43 fnfvima 7178 . . 3 ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴 (𝐹 “ ran 𝐺) ∈ (𝐹 “ ran 𝐺)) → (𝐹 (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)))
445, 9, 42, 43syl3anc 1372 . 2 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)))
45 simpl 484 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → 𝐴 ∈ FinIII)
466, 7sstrid 3954 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴)
47 incom 4160 . . . . . . . . 9 (dom 𝐹 ∩ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹)
48 frn 6671 . . . . . . . . . . . 12 (𝐺:ω⟶𝒫 𝐴 → ran 𝐺 ⊆ 𝒫 𝐴)
4948adantl 483 . . . . . . . . . . 11 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ⊆ 𝒫 𝐴)
503fdmd 6675 . . . . . . . . . . 11 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → dom 𝐹 = 𝒫 𝐴)
5149, 50sseqtrrd 3984 . . . . . . . . . 10 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ⊆ dom 𝐹)
52 df-ss 3926 . . . . . . . . . 10 (ran 𝐺 ⊆ dom 𝐹 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
5351, 52sylib 217 . . . . . . . . 9 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
5447, 53eqtrid 2790 . . . . . . . 8 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) = ran 𝐺)
55 fdm 6673 . . . . . . . . . . 11 (𝐺:ω⟶𝒫 𝐴 → dom 𝐺 = ω)
5655adantl 483 . . . . . . . . . 10 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → dom 𝐺 = ω)
57 peano1 7816 . . . . . . . . . . 11 ∅ ∈ ω
58 ne0i 4293 . . . . . . . . . . 11 (∅ ∈ ω → ω ≠ ∅)
5957, 58mp1i 13 . . . . . . . . . 10 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ω ≠ ∅)
6056, 59eqnetrd 3010 . . . . . . . . 9 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → dom 𝐺 ≠ ∅)
61 dm0rn0 5877 . . . . . . . . . 10 (dom 𝐺 = ∅ ↔ ran 𝐺 = ∅)
6261necon3bii 2995 . . . . . . . . 9 (dom 𝐺 ≠ ∅ ↔ ran 𝐺 ≠ ∅)
6360, 62sylib 217 . . . . . . . 8 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ≠ ∅)
6454, 63eqnetrd 3010 . . . . . . 7 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) ≠ ∅)
65 imadisj 6029 . . . . . . . 8 ((𝐹 “ ran 𝐺) = ∅ ↔ (dom 𝐹 ∩ ran 𝐺) = ∅)
6665necon3bii 2995 . . . . . . 7 ((𝐹 “ ran 𝐺) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝐺) ≠ ∅)
6764, 66sylibr 233 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 “ ran 𝐺) ≠ ∅)
681isf34lem5 10248 . . . . . 6 ((𝐴 ∈ FinIII ∧ ((𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴 ∧ (𝐹 “ ran 𝐺) ≠ ∅)) → (𝐹 (𝐹 “ ran 𝐺)) = (𝐹 “ (𝐹 “ ran 𝐺)))
6945, 46, 67, 68syl12anc 836 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 (𝐹 “ ran 𝐺)) = (𝐹 “ (𝐹 “ ran 𝐺)))
701isf34lem3 10245 . . . . . . 7 ((𝐴 ∈ FinIII ∧ ran 𝐺 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
7145, 49, 70syl2anc 585 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
7271unieqd 4878 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
7369, 72eqtrd 2778 . . . 4 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 (𝐹 “ ran 𝐺)) = ran 𝐺)
7473, 71eleq12d 2833 . . 3 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ((𝐹 (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)) ↔ ran 𝐺 ∈ ran 𝐺))
75743adant3 1133 . 2 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ((𝐹 (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)) ↔ ran 𝐺 ∈ ran 𝐺))
7644, 75mpbid 231 1 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ran 𝐺 ∈ ran 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2942  wral 3063  cdif 3906  cin 3908  wss 3909  c0 4281  𝒫 cpw 4559   cuni 4864   cint 4906  cmpt 5187  dom cdm 5631  ran crn 5632  cima 5634  ccom 5635  suc csuc 6316   Fn wfn 6487  wf 6488  cfv 6492  ωcom 7793  FinIIIcfin3 10151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6250  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7306  df-ov 7353  df-rpss 7651  df-om 7794  df-2nd 7913  df-frecs 8180  df-wrecs 8211  df-recs 8285  df-rdg 8324  df-1o 8380  df-er 8582  df-en 8818  df-dom 8819  df-sdom 8820  df-fin 8821  df-wdom 9435  df-card 9809  df-fin4 10157  df-fin3 10158
This theorem is referenced by:  isf34lem6  10250  fin34i  10251
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