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Theorem isf34lem6 9455
Description: Lemma for isfin3-4 9457. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Hypothesis
Ref Expression
compss.a 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
Assertion
Ref Expression
isf34lem6 (𝐴𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓)))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝑓,𝐹,𝑦   𝑥,𝑉,𝑦
Allowed substitution hints:   𝐹(𝑥)   𝑉(𝑓)

Proof of Theorem isf34lem6
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 elmapi 8082 . . . 4 (𝑓 ∈ (𝒫 𝐴𝑚 ω) → 𝑓:ω⟶𝒫 𝐴)
2 compss.a . . . . . 6 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
32isf34lem7 9454 . . . . 5 ((𝐴 ∈ FinIII𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦)) → ran 𝑓 ∈ ran 𝑓)
433expia 1150 . . . 4 ((𝐴 ∈ FinIII𝑓:ω⟶𝒫 𝐴) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
51, 4sylan2 586 . . 3 ((𝐴 ∈ FinIII𝑓 ∈ (𝒫 𝐴𝑚 ω)) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
65ralrimiva 3113 . 2 (𝐴 ∈ FinIII → ∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
7 elmapex 8081 . . . . . . . . . . 11 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝒫 𝐴 ∈ V ∧ ω ∈ V))
87simpld 488 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝒫 𝐴 ∈ V)
9 pwexb 7173 . . . . . . . . . 10 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
108, 9sylibr 225 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝐴 ∈ V)
112isf34lem2 9448 . . . . . . . . 9 (𝐴 ∈ V → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
1210, 11syl 17 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
13 elmapi 8082 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝑔:ω⟶𝒫 𝐴)
14 fco 6240 . . . . . . . 8 ((𝐹:𝒫 𝐴⟶𝒫 𝐴𝑔:ω⟶𝒫 𝐴) → (𝐹𝑔):ω⟶𝒫 𝐴)
1512, 13, 14syl2anc 579 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹𝑔):ω⟶𝒫 𝐴)
16 elmapg 8073 . . . . . . . 8 ((𝒫 𝐴 ∈ V ∧ ω ∈ V) → ((𝐹𝑔) ∈ (𝒫 𝐴𝑚 ω) ↔ (𝐹𝑔):ω⟶𝒫 𝐴))
177, 16syl 17 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ((𝐹𝑔) ∈ (𝒫 𝐴𝑚 ω) ↔ (𝐹𝑔):ω⟶𝒫 𝐴))
1815, 17mpbird 248 . . . . . 6 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹𝑔) ∈ (𝒫 𝐴𝑚 ω))
19 fveq1 6374 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓𝑦) = ((𝐹𝑔)‘𝑦))
20 fveq1 6374 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓‘suc 𝑦) = ((𝐹𝑔)‘suc 𝑦))
2119, 20sseq12d 3794 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ((𝑓𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
2221ralbidv 3133 . . . . . . . 8 (𝑓 = (𝐹𝑔) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
23 rneq 5519 . . . . . . . . . . 11 (𝑓 = (𝐹𝑔) → ran 𝑓 = ran (𝐹𝑔))
24 rnco2 5828 . . . . . . . . . . 11 ran (𝐹𝑔) = (𝐹 “ ran 𝑔)
2523, 24syl6eq 2815 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
2625unieqd 4604 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
2726, 25eleq12d 2838 . . . . . . . 8 (𝑓 = (𝐹𝑔) → ( ran 𝑓 ∈ ran 𝑓 (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)))
2822, 27imbi12d 335 . . . . . . 7 (𝑓 = (𝐹𝑔) → ((∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓) ↔ (∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦) → (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔))))
2928rspccv 3458 . . . . . 6 (∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓) → ((𝐹𝑔) ∈ (𝒫 𝐴𝑚 ω) → (∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦) → (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔))))
3018, 29syl5 34 . . . . 5 (∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓) → (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦) → (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔))))
31 sscon 3906 . . . . . . . . 9 ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → (𝐴 ∖ (𝑔𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦)))
3210adantr 472 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → 𝐴 ∈ V)
3313ffvelrnda 6549 . . . . . . . . . . . 12 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔𝑦) ∈ 𝒫 𝐴)
3433elpwid 4327 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔𝑦) ⊆ 𝐴)
352isf34lem1 9447 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑔𝑦) ⊆ 𝐴) → (𝐹‘(𝑔𝑦)) = (𝐴 ∖ (𝑔𝑦)))
3632, 34, 35syl2anc 579 . . . . . . . . . 10 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔𝑦)) = (𝐴 ∖ (𝑔𝑦)))
37 peano2 7284 . . . . . . . . . . . . 13 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
38 ffvelrn 6547 . . . . . . . . . . . . 13 ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
3913, 37, 38syl2an 589 . . . . . . . . . . . 12 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
4039elpwid 4327 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ⊆ 𝐴)
412isf34lem1 9447 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑔‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
4232, 40, 41syl2anc 579 . . . . . . . . . 10 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
4336, 42sseq12d 3794 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝐹‘(𝑔𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦)) ↔ (𝐴 ∖ (𝑔𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦))))
4431, 43syl5ibr 237 . . . . . . . 8 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → (𝐹‘(𝑔𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
45 fvco3 6464 . . . . . . . . . 10 ((𝑔:ω⟶𝒫 𝐴𝑦 ∈ ω) → ((𝐹𝑔)‘𝑦) = (𝐹‘(𝑔𝑦)))
4613, 45sylan 575 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝐹𝑔)‘𝑦) = (𝐹‘(𝑔𝑦)))
47 fvco3 6464 . . . . . . . . . 10 ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → ((𝐹𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
4813, 37, 47syl2an 589 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝐹𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
4946, 48sseq12d 3794 . . . . . . . 8 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦) ↔ (𝐹‘(𝑔𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
5044, 49sylibrd 250 . . . . . . 7 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
5150ralimdva 3109 . . . . . 6 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
5212ffnd 6224 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝐹 Fn 𝒫 𝐴)
53 imassrn 5659 . . . . . . . . 9 (𝐹 “ ran 𝑔) ⊆ ran 𝐹
5412frnd 6230 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝐹 ⊆ 𝒫 𝐴)
5553, 54syl5ss 3772 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴)
56 fnfvima 6689 . . . . . . . . 9 ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴 (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)) → (𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔)))
57563expia 1150 . . . . . . . 8 ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴) → ( (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → (𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔))))
5852, 55, 57syl2anc 579 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ( (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → (𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔))))
59 incom 3967 . . . . . . . . . . . . 13 (dom 𝐹 ∩ ran 𝑔) = (ran 𝑔 ∩ dom 𝐹)
6013frnd 6230 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝑔 ⊆ 𝒫 𝐴)
6112fdmd 6232 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → dom 𝐹 = 𝒫 𝐴)
6260, 61sseqtr4d 3802 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝑔 ⊆ dom 𝐹)
63 df-ss 3746 . . . . . . . . . . . . . 14 (ran 𝑔 ⊆ dom 𝐹 ↔ (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
6462, 63sylib 209 . . . . . . . . . . . . 13 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
6559, 64syl5eq 2811 . . . . . . . . . . . 12 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (dom 𝐹 ∩ ran 𝑔) = ran 𝑔)
6613fdmd 6232 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → dom 𝑔 = ω)
67 peano1 7283 . . . . . . . . . . . . . . 15 ∅ ∈ ω
68 ne0i 4085 . . . . . . . . . . . . . . 15 (∅ ∈ ω → ω ≠ ∅)
6967, 68mp1i 13 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ω ≠ ∅)
7066, 69eqnetrd 3004 . . . . . . . . . . . . 13 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → dom 𝑔 ≠ ∅)
71 dm0rn0 5510 . . . . . . . . . . . . . 14 (dom 𝑔 = ∅ ↔ ran 𝑔 = ∅)
7271necon3bii 2989 . . . . . . . . . . . . 13 (dom 𝑔 ≠ ∅ ↔ ran 𝑔 ≠ ∅)
7370, 72sylib 209 . . . . . . . . . . . 12 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝑔 ≠ ∅)
7465, 73eqnetrd 3004 . . . . . . . . . . 11 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
75 imadisj 5666 . . . . . . . . . . . 12 ((𝐹 “ ran 𝑔) = ∅ ↔ (dom 𝐹 ∩ ran 𝑔) = ∅)
7675necon3bii 2989 . . . . . . . . . . 11 ((𝐹 “ ran 𝑔) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
7774, 76sylibr 225 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ ran 𝑔) ≠ ∅)
782isf34lem4 9452 . . . . . . . . . 10 ((𝐴 ∈ V ∧ ((𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ≠ ∅)) → (𝐹 (𝐹 “ ran 𝑔)) = (𝐹 “ (𝐹 “ ran 𝑔)))
7910, 55, 77, 78syl12anc 865 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 (𝐹 “ ran 𝑔)) = (𝐹 “ (𝐹 “ ran 𝑔)))
802isf34lem3 9450 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ ran 𝑔 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8110, 60, 80syl2anc 579 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8281inteqd 4638 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8379, 82eqtrd 2799 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 (𝐹 “ ran 𝑔)) = ran 𝑔)
8483, 81eleq12d 2838 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ((𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔)) ↔ ran 𝑔 ∈ ran 𝑔))
8558, 84sylibd 230 . . . . . 6 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ( (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → ran 𝑔 ∈ ran 𝑔))
8651, 85imim12d 81 . . . . 5 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ((∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦) → (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)) → (∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ran 𝑔 ∈ ran 𝑔)))
8730, 86sylcom 30 . . . 4 (∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓) → (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ran 𝑔 ∈ ran 𝑔)))
8887ralrimiv 3112 . . 3 (∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓) → ∀𝑔 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ran 𝑔 ∈ ran 𝑔))
89 isfin3-3 9443 . . 3 (𝐴𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑔 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ran 𝑔 ∈ ran 𝑔)))
9088, 89syl5ibr 237 . 2 (𝐴𝑉 → (∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓) → 𝐴 ∈ FinIII))
916, 90impbid2 217 1 (𝐴𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  Vcvv 3350  cdif 3729  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315   cuni 4594   cint 4633  cmpt 4888  dom cdm 5277  ran crn 5278  cima 5280  ccom 5281  suc csuc 5910   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  ωcom 7263  𝑚 cmap 8060  FinIIIcfin3 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-rpss 7135  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-seqom 7747  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-wdom 8671  df-card 9016  df-fin4 9362  df-fin3 9363
This theorem is referenced by:  isfin3-4  9457
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