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Theorem isf34lem7 9145
Description: Lemma for isfin3-4 9148. (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 9139 . . . . . 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 6002 . . . 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 5436 . . . 4 (𝐹 “ ran 𝐺) ⊆ ran 𝐹
8 frn 6010 . . . . . 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 3594 . . 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 6015 . . . . . . 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 3722 . . . . . . . 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 7033 . . . . . . . . . . 11 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
19 fvco3 6232 . . . . . . . . . . 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 6313 . . . . . . . . . . . . 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 4141 . . . . . . . . . . 11 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ⊆ 𝐴)
251isf34lem1 9138 . . . . . . . . . . 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 2655 . . . . . . . . 9 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹𝐺)‘suc 𝑦) = (𝐴 ∖ (𝐺‘suc 𝑦)))
28 fvco3 6232 . . . . . . . . . . 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 6313 . . . . . . . . . . . . 13 ((𝐺:ω⟶𝒫 𝐴𝑦 ∈ ω) → (𝐺𝑦) ∈ 𝒫 𝐴)
3130adantll 749 . . . . . . . . . . . 12 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺𝑦) ∈ 𝒫 𝐴)
3231elpwid 4141 . . . . . . . . . . 11 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺𝑦) ⊆ 𝐴)
331isf34lem1 9138 . . . . . . . . . . 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 2655 . . . . . . . . 9 (((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹𝐺)‘𝑦) = (𝐴 ∖ (𝐺𝑦)))
3627, 35sseq12d 3613 . . . . . . . 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 2956 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦) → ∀𝑦 ∈ ω ((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦)))
39383impia 1258 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ∀𝑦 ∈ ω ((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦))
40 fin33i 9135 . . . . 5 ((𝐴 ∈ FinIII ∧ (𝐹𝐺):ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω ((𝐹𝐺)‘suc 𝑦) ⊆ ((𝐹𝐺)‘𝑦)) → ran (𝐹𝐺) ∈ ran (𝐹𝐺))
4112, 15, 39, 40syl3anc 1323 . . . 4 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ran (𝐹𝐺) ∈ ran (𝐹𝐺))
42 rnco2 5601 . . . . 5 ran (𝐹𝐺) = (𝐹 “ ran 𝐺)
4342inteqi 4444 . . . 4 ran (𝐹𝐺) = (𝐹 “ ran 𝐺)
4441, 43, 423eltr3g 2714 . . 3 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 “ ran 𝐺) ∈ (𝐹 “ ran 𝐺))
45 fnfvima 6450 . . 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 1323 . 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 3594 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴)
49 incom 3783 . . . . . . . . 9 (dom 𝐹 ∩ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹)
50 frn 6010 . . . . . . . . . . . 12 (𝐺:ω⟶𝒫 𝐴 → ran 𝐺 ⊆ 𝒫 𝐴)
5150adantl 482 . . . . . . . . . . 11 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ⊆ 𝒫 𝐴)
52 fdm 6008 . . . . . . . . . . . 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 3621 . . . . . . . . . 10 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ⊆ dom 𝐹)
55 df-ss 3569 . . . . . . . . . 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 2667 . . . . . . . 8 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) = ran 𝐺)
58 fdm 6008 . . . . . . . . . . 11 (𝐺:ω⟶𝒫 𝐴 → dom 𝐺 = ω)
5958adantl 482 . . . . . . . . . 10 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → dom 𝐺 = ω)
60 peano1 7032 . . . . . . . . . . 11 ∅ ∈ ω
61 ne0i 3897 . . . . . . . . . . 11 (∅ ∈ ω → ω ≠ ∅)
6260, 61mp1i 13 . . . . . . . . . 10 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ω ≠ ∅)
6359, 62eqnetrd 2857 . . . . . . . . 9 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → dom 𝐺 ≠ ∅)
64 dm0rn0 5302 . . . . . . . . . 10 (dom 𝐺 = ∅ ↔ ran 𝐺 = ∅)
6564necon3bii 2842 . . . . . . . . 9 (dom 𝐺 ≠ ∅ ↔ ran 𝐺 ≠ ∅)
6663, 65sylib 208 . . . . . . . 8 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ≠ ∅)
6757, 66eqnetrd 2857 . . . . . . 7 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) ≠ ∅)
68 imadisj 5443 . . . . . . . 8 ((𝐹 “ ran 𝐺) = ∅ ↔ (dom 𝐹 ∩ ran 𝐺) = ∅)
6968necon3bii 2842 . . . . . . 7 ((𝐹 “ ran 𝐺) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝐺) ≠ ∅)
7067, 69sylibr 224 . . . . . 6 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 “ ran 𝐺) ≠ ∅)
711isf34lem5 9144 . . . . . 6 ((𝐴 ∈ FinIII ∧ ((𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴 ∧ (𝐹 “ ran 𝐺) ≠ ∅)) → (𝐹 (𝐹 “ ran 𝐺)) = (𝐹 “ (𝐹 “ ran 𝐺)))
7247, 48, 70, 71syl12anc 1321 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 (𝐹 “ ran 𝐺)) = (𝐹 “ (𝐹 “ ran 𝐺)))
731isf34lem3 9141 . . . . . . 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 4412 . . . . 5 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
7672, 75eqtrd 2655 . . . 4 ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴) → (𝐹 (𝐹 “ ran 𝐺)) = ran 𝐺)
7776, 74eleq12d 2692 . . 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 1480  wcel 1987  wne 2790  wral 2907  cdif 3552  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130   cuni 4402   cint 4440  cmpt 4673  dom cdm 5074  ran crn 5075  cima 5077  ccom 5078  suc csuc 5684   Fn wfn 5842  wf 5843  cfv 5847  ωcom 7012  FinIIIcfin3 9047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-rpss 6890  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-wdom 8408  df-card 8709  df-fin4 9053  df-fin3 9054
This theorem is referenced by:  isf34lem6  9146  fin34i  9147
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