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Theorem isf34lem7 9402
Description: Lemma for isfin3-4 9405. (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 9396 . . . . . 6 (𝐴 ∈ FinIII𝐹:𝒫 𝐴⟶𝒫 𝐴)
32adantr 466 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
433adant3 1125 . . . 4 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
5 ffn 6185 . . . 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 5618 . . . 4 (𝐹 “ ran 𝐺) ⊆ ran 𝐹
8 frn 6193 . . . . . 6 (𝐹:𝒫 𝐴⟶𝒫 𝐴 → ran 𝐹 ⊆ 𝒫 𝐴)
93, 8syl 17 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ran 𝐹 ⊆ 𝒫 𝐴)
1093adant3 1125 . . . 4 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ran 𝐹 ⊆ 𝒫 𝐴)
117, 10syl5ss 3761 . . 3 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴)
12 simp1 1129 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐴 ∈ FinIII)
13 fco 6198 . . . . . . 7 ((𝐹:𝒫 𝐴⟶𝒫 𝐴𝐺:ω⟶𝒫 𝐴) → (𝐹𝐺):ω⟶𝒫 𝐴)
142, 13sylan 561 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹𝐺):ω⟶𝒫 𝐴)
15143adant3 1125 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹𝐺):ω⟶𝒫 𝐴)
16 sscon 3893 . . . . . . . 8 ((𝐺𝑦) ⊆ (𝐺‘suc 𝑦) → (𝐴 ∖ (𝐺‘suc 𝑦)) ⊆ (𝐴 ∖ (𝐺𝑦)))
17 simpr 471 . . . . . . . . . . 11 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → 𝐺:ω⟶𝒫 𝐴)
18 peano2 7232 . . . . . . . . . . 11 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
19 fvco3 6417 . . . . . . . . . . 11 ((𝐺:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → ((𝐹𝐺)‘suc 𝑦) = (𝐹‘(𝐺‘suc 𝑦)))
2017, 18, 19syl2an 575 . . . . . . . . . 10 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹𝐺)‘suc 𝑦) = (𝐹‘(𝐺‘suc 𝑦)))
21 simpll 742 . . . . . . . . . . 11 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → 𝐴 ∈ FinIII)
22 ffvelrn 6500 . . . . . . . . . . . . 13 ((𝐺:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ∈ 𝒫 𝐴)
2317, 18, 22syl2an 575 . . . . . . . . . . . 12 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ∈ 𝒫 𝐴)
2423elpwid 4307 . . . . . . . . . . 11 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ⊆ 𝐴)
251isf34lem1 9395 . . . . . . . . . . 11 ((𝐴 ∈ FinIII ∧ (𝐺‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝐺‘suc 𝑦)) = (𝐴 ∖ (𝐺‘suc 𝑦)))
2621, 24, 25syl2anc 565 . . . . . . . . . 10 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐹‘(𝐺‘suc 𝑦)) = (𝐴 ∖ (𝐺‘suc 𝑦)))
2720, 26eqtrd 2804 . . . . . . . . 9 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹𝐺)‘suc 𝑦) = (𝐴 ∖ (𝐺‘suc 𝑦)))
28 fvco3 6417 . . . . . . . . . . 11 ((𝐺:ω⟶𝒫 𝐴𝑦 ∈ ω) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
2928adantll 685 . . . . . . . . . 10 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
30 ffvelrn 6500 . . . . . . . . . . . . 13 ((𝐺:ω⟶𝒫 𝐴𝑦 ∈ ω) → (𝐺𝑦) ∈ 𝒫 𝐴)
3130adantll 685 . . . . . . . . . . . 12 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺𝑦) ∈ 𝒫 𝐴)
3231elpwid 4307 . . . . . . . . . . 11 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺𝑦) ⊆ 𝐴)
331isf34lem1 9395 . . . . . . . . . . 11 ((𝐴 ∈ FinIII ∧ (𝐺𝑦) ⊆ 𝐴) → (𝐹‘(𝐺𝑦)) = (𝐴 ∖ (𝐺𝑦)))
3421, 32, 33syl2anc 565 . . . . . . . . . 10 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐹‘(𝐺𝑦)) = (𝐴 ∖ (𝐺𝑦)))
3529, 34eqtrd 2804 . . . . . . . . 9 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹𝐺)‘𝑦) = (𝐴 ∖ (𝐺𝑦)))
3627, 35sseq12d 3781 . . . . . . . 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 3110 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦) → ∀𝑦 ∈ ω ((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦)))
39383impia 1108 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ∀𝑦 ∈ ω ((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦))
40 fin33i 9392 . . . . 5 ((𝐴 ∈ FinIII ∧ (𝐹𝐺):ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω ((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦)) → ran (𝐹𝐺) ∈ ran (𝐹𝐺))
4112, 15, 39, 40syl3anc 1475 . . . 4 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ran (𝐹𝐺) ∈ ran (𝐹𝐺))
42 rnco2 5786 . . . . 5 ran (𝐹𝐺) = (𝐹 “ ran 𝐺)
4342inteqi 4613 . . . 4 ran (𝐹𝐺) = (𝐹 “ ran 𝐺)
4441, 43, 423eltr3g 2865 . . 3 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 “ ran 𝐺) ∈ (𝐹 “ ran 𝐺))
45 fnfvima 6638 . . 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 1475 . 2 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)))
47 simpl 468 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → 𝐴 ∈ FinIII)
487, 9syl5ss 3761 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴)
49 incom 3954 . . . . . . . . 9 (dom 𝐹 ∩ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹)
50 frn 6193 . . . . . . . . . . . 12 (𝐺:ω⟶𝒫 𝐴 → ran 𝐺 ⊆ 𝒫 𝐴)
5150adantl 467 . . . . . . . . . . 11 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ⊆ 𝒫 𝐴)
52 fdm 6191 . . . . . . . . . . . 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 3789 . . . . . . . . . 10 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ⊆ dom 𝐹)
55 df-ss 3735 . . . . . . . . . 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 2816 . . . . . . . 8 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) = ran 𝐺)
58 fdm 6191 . . . . . . . . . . 11 (𝐺:ω⟶𝒫 𝐴 → dom 𝐺 = ω)
5958adantl 467 . . . . . . . . . 10 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → dom 𝐺 = ω)
60 peano1 7231 . . . . . . . . . . 11 ∅ ∈ ω
61 ne0i 4067 . . . . . . . . . . 11 (∅ ∈ ω → ω ≠ ∅)
6260, 61mp1i 13 . . . . . . . . . 10 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ω ≠ ∅)
6359, 62eqnetrd 3009 . . . . . . . . 9 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → dom 𝐺 ≠ ∅)
64 dm0rn0 5480 . . . . . . . . . 10 (dom 𝐺 = ∅ ↔ ran 𝐺 = ∅)
6564necon3bii 2994 . . . . . . . . 9 (dom 𝐺 ≠ ∅ ↔ ran 𝐺 ≠ ∅)
6663, 65sylib 208 . . . . . . . 8 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ≠ ∅)
6757, 66eqnetrd 3009 . . . . . . 7 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) ≠ ∅)
68 imadisj 5625 . . . . . . . 8 ((𝐹 “ ran 𝐺) = ∅ ↔ (dom 𝐹 ∩ ran 𝐺) = ∅)
6968necon3bii 2994 . . . . . . 7 ((𝐹 “ ran 𝐺) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝐺) ≠ ∅)
7067, 69sylibr 224 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 “ ran 𝐺) ≠ ∅)
711isf34lem5 9401 . . . . . 6 ((𝐴 ∈ FinIII ∧ ((𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴 ∧ (𝐹 “ ran 𝐺) ≠ ∅)) → (𝐹 (𝐹 “ ran 𝐺)) = (𝐹 “ (𝐹 “ ran 𝐺)))
7247, 48, 70, 71syl12anc 1473 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 (𝐹 “ ran 𝐺)) = (𝐹 “ (𝐹 “ ran 𝐺)))
731isf34lem3 9398 . . . . . . 7 ((𝐴 ∈ FinIII ∧ ran 𝐺 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
7447, 51, 73syl2anc 565 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
7574unieqd 4582 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
7672, 75eqtrd 2804 . . . 4 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 (𝐹 “ ran 𝐺)) = ran 𝐺)
7776, 74eleq12d 2843 . . 3 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ((𝐹 (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)) ↔ ran 𝐺 ∈ ran 𝐺))
78773adant3 1125 . 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 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wral 3060  cdif 3718  cin 3720  wss 3721  c0 4061  𝒫 cpw 4295   cuni 4572   cint 4609  cmpt 4861  dom cdm 5249  ran crn 5250  cima 5252  ccom 5253  suc csuc 5868   Fn wfn 6026  wf 6027  cfv 6031  ωcom 7211  FinIIIcfin3 9304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  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-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-rpss 7083  df-om 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-wdom 8619  df-card 8964  df-fin4 9310  df-fin3 9311
This theorem is referenced by:  isf34lem6  9403  fin34i  9404
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