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Theorem isf34lem7 9186
Description: Lemma for isfin3-4 9189. (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 9180 . . . . . 6 (𝐴 ∈ FinIII𝐹:𝒫 𝐴⟶𝒫 𝐴)
32adantr 481 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
433adant3 1079 . . . 4 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
5 ffn 6032 . . . 4 (𝐹:𝒫 𝐴⟶𝒫 𝐴𝐹 Fn 𝒫 𝐴)
64, 5syl 17 . . 3 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐹 Fn 𝒫 𝐴)
7 imassrn 5465 . . . 4 (𝐹 “ ran 𝐺) ⊆ ran 𝐹
8 frn 6040 . . . . . 6 (𝐹:𝒫 𝐴⟶𝒫 𝐴 → ran 𝐹 ⊆ 𝒫 𝐴)
93, 8syl 17 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ran 𝐹 ⊆ 𝒫 𝐴)
1093adant3 1079 . . . 4 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ran 𝐹 ⊆ 𝒫 𝐴)
117, 10syl5ss 3606 . . 3 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴)
12 simp1 1059 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐴 ∈ FinIII)
13 fco 6045 . . . . . . 7 ((𝐹:𝒫 𝐴⟶𝒫 𝐴𝐺:ω⟶𝒫 𝐴) → (𝐹𝐺):ω⟶𝒫 𝐴)
142, 13sylan 488 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹𝐺):ω⟶𝒫 𝐴)
15143adant3 1079 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹𝐺):ω⟶𝒫 𝐴)
16 sscon 3736 . . . . . . . 8 ((𝐺𝑦) ⊆ (𝐺‘suc 𝑦) → (𝐴 ∖ (𝐺‘suc 𝑦)) ⊆ (𝐴 ∖ (𝐺𝑦)))
17 simpr 477 . . . . . . . . . . 11 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → 𝐺:ω⟶𝒫 𝐴)
18 peano2 7071 . . . . . . . . . . 11 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
19 fvco3 6262 . . . . . . . . . . 11 ((𝐺:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → ((𝐹𝐺)‘suc 𝑦) = (𝐹‘(𝐺‘suc 𝑦)))
2017, 18, 19syl2an 494 . . . . . . . . . 10 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹𝐺)‘suc 𝑦) = (𝐹‘(𝐺‘suc 𝑦)))
21 simpll 789 . . . . . . . . . . 11 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → 𝐴 ∈ FinIII)
22 ffvelrn 6343 . . . . . . . . . . . . 13 ((𝐺:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ∈ 𝒫 𝐴)
2317, 18, 22syl2an 494 . . . . . . . . . . . 12 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ∈ 𝒫 𝐴)
2423elpwid 4161 . . . . . . . . . . 11 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ⊆ 𝐴)
251isf34lem1 9179 . . . . . . . . . . 11 ((𝐴 ∈ FinIII ∧ (𝐺‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝐺‘suc 𝑦)) = (𝐴 ∖ (𝐺‘suc 𝑦)))
2621, 24, 25syl2anc 692 . . . . . . . . . 10 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐹‘(𝐺‘suc 𝑦)) = (𝐴 ∖ (𝐺‘suc 𝑦)))
2720, 26eqtrd 2654 . . . . . . . . 9 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹𝐺)‘suc 𝑦) = (𝐴 ∖ (𝐺‘suc 𝑦)))
28 fvco3 6262 . . . . . . . . . . 11 ((𝐺:ω⟶𝒫 𝐴𝑦 ∈ ω) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
2928adantll 749 . . . . . . . . . 10 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
30 ffvelrn 6343 . . . . . . . . . . . . 13 ((𝐺:ω⟶𝒫 𝐴𝑦 ∈ ω) → (𝐺𝑦) ∈ 𝒫 𝐴)
3130adantll 749 . . . . . . . . . . . 12 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺𝑦) ∈ 𝒫 𝐴)
3231elpwid 4161 . . . . . . . . . . 11 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺𝑦) ⊆ 𝐴)
331isf34lem1 9179 . . . . . . . . . . 11 ((𝐴 ∈ FinIII ∧ (𝐺𝑦) ⊆ 𝐴) → (𝐹‘(𝐺𝑦)) = (𝐴 ∖ (𝐺𝑦)))
3421, 32, 33syl2anc 692 . . . . . . . . . 10 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐹‘(𝐺𝑦)) = (𝐴 ∖ (𝐺𝑦)))
3529, 34eqtrd 2654 . . . . . . . . 9 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹𝐺)‘𝑦) = (𝐴 ∖ (𝐺𝑦)))
3627, 35sseq12d 3626 . . . . . . . 8 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦) ↔ (𝐴 ∖ (𝐺‘suc 𝑦)) ⊆ (𝐴 ∖ (𝐺𝑦))))
3716, 36syl5ibr 236 . . . . . . 7 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐺𝑦) ⊆ (𝐺‘suc 𝑦) → ((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦)))
3837ralimdva 2959 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦) → ∀𝑦 ∈ ω ((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦)))
39383impia 1259 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ∀𝑦 ∈ ω ((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦))
40 fin33i 9176 . . . . 5 ((𝐴 ∈ FinIII ∧ (𝐹𝐺):ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω ((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦)) → ran (𝐹𝐺) ∈ ran (𝐹𝐺))
4112, 15, 39, 40syl3anc 1324 . . . 4 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ran (𝐹𝐺) ∈ ran (𝐹𝐺))
42 rnco2 5630 . . . . 5 ran (𝐹𝐺) = (𝐹 “ ran 𝐺)
4342inteqi 4470 . . . 4 ran (𝐹𝐺) = (𝐹 “ ran 𝐺)
4441, 43, 423eltr3g 2715 . . 3 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 “ ran 𝐺) ∈ (𝐹 “ ran 𝐺))
45 fnfvima 6481 . . 3 ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴 (𝐹 “ ran 𝐺) ∈ (𝐹 “ ran 𝐺)) → (𝐹 (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)))
466, 11, 44, 45syl3anc 1324 . 2 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)))
47 simpl 473 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → 𝐴 ∈ FinIII)
487, 9syl5ss 3606 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴)
49 incom 3797 . . . . . . . . 9 (dom 𝐹 ∩ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹)
50 frn 6040 . . . . . . . . . . . 12 (𝐺:ω⟶𝒫 𝐴 → ran 𝐺 ⊆ 𝒫 𝐴)
5150adantl 482 . . . . . . . . . . 11 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ⊆ 𝒫 𝐴)
52 fdm 6038 . . . . . . . . . . . 12 (𝐹:𝒫 𝐴⟶𝒫 𝐴 → dom 𝐹 = 𝒫 𝐴)
533, 52syl 17 . . . . . . . . . . 11 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → dom 𝐹 = 𝒫 𝐴)
5451, 53sseqtr4d 3634 . . . . . . . . . 10 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ⊆ dom 𝐹)
55 df-ss 3581 . . . . . . . . . 10 (ran 𝐺 ⊆ dom 𝐹 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
5654, 55sylib 208 . . . . . . . . 9 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
5749, 56syl5eq 2666 . . . . . . . 8 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) = ran 𝐺)
58 fdm 6038 . . . . . . . . . . 11 (𝐺:ω⟶𝒫 𝐴 → dom 𝐺 = ω)
5958adantl 482 . . . . . . . . . 10 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → dom 𝐺 = ω)
60 peano1 7070 . . . . . . . . . . 11 ∅ ∈ ω
61 ne0i 3913 . . . . . . . . . . 11 (∅ ∈ ω → ω ≠ ∅)
6260, 61mp1i 13 . . . . . . . . . 10 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ω ≠ ∅)
6359, 62eqnetrd 2858 . . . . . . . . 9 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → dom 𝐺 ≠ ∅)
64 dm0rn0 5331 . . . . . . . . . 10 (dom 𝐺 = ∅ ↔ ran 𝐺 = ∅)
6564necon3bii 2843 . . . . . . . . 9 (dom 𝐺 ≠ ∅ ↔ ran 𝐺 ≠ ∅)
6663, 65sylib 208 . . . . . . . 8 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ≠ ∅)
6757, 66eqnetrd 2858 . . . . . . 7 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) ≠ ∅)
68 imadisj 5472 . . . . . . . 8 ((𝐹 “ ran 𝐺) = ∅ ↔ (dom 𝐹 ∩ ran 𝐺) = ∅)
6968necon3bii 2843 . . . . . . 7 ((𝐹 “ ran 𝐺) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝐺) ≠ ∅)
7067, 69sylibr 224 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 “ ran 𝐺) ≠ ∅)
711isf34lem5 9185 . . . . . 6 ((𝐴 ∈ FinIII ∧ ((𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴 ∧ (𝐹 “ ran 𝐺) ≠ ∅)) → (𝐹 (𝐹 “ ran 𝐺)) = (𝐹 “ (𝐹 “ ran 𝐺)))
7247, 48, 70, 71syl12anc 1322 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 (𝐹 “ ran 𝐺)) = (𝐹 “ (𝐹 “ ran 𝐺)))
731isf34lem3 9182 . . . . . . 7 ((𝐴 ∈ FinIII ∧ ran 𝐺 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
7447, 51, 73syl2anc 692 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
7574unieqd 4437 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
7672, 75eqtrd 2654 . . . 4 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 (𝐹 “ ran 𝐺)) = ran 𝐺)
7776, 74eleq12d 2693 . . 3 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ((𝐹 (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)) ↔ ran 𝐺 ∈ ran 𝐺))
78773adant3 1079 . 2 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ((𝐹 (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)) ↔ ran 𝐺 ∈ ran 𝐺))
7946, 78mpbid 222 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 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791  wral 2909  cdif 3564  cin 3566  wss 3567  c0 3907  𝒫 cpw 4149   cuni 4427   cint 4466  cmpt 4720  dom cdm 5104  ran crn 5105  cima 5107  ccom 5108  suc csuc 5713   Fn wfn 5871  wf 5872  cfv 5876  ωcom 7050  FinIIIcfin3 9088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-rpss 6922  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-wdom 8449  df-card 8750  df-fin4 9094  df-fin3 9095
This theorem is referenced by:  isf34lem6  9187  fin34i  9188
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