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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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