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Theorem ttukeylem5 10554
Description: Lemma for ttukey 10559. The 𝐺 function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
ttukeylem.1 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
ttukeylem.2 (𝜑𝐵𝐴)
ttukeylem.3 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
ttukeylem.4 𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))
Assertion
Ref Expression
ttukeylem5 ((𝜑 ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝐶𝐷)) → (𝐺𝐶) ⊆ (𝐺𝐷))
Distinct variable groups:   𝑥,𝑧,𝐶   𝑥,𝐷   𝑥,𝐺,𝑧   𝜑,𝑧   𝑥,𝐴,𝑧   𝑥,𝐵,𝑧   𝑥,𝐹,𝑧
Allowed substitution hints:   𝜑(𝑥)   𝐷(𝑧)

Proof of Theorem ttukeylem5
Dummy variables 𝑎 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq2 4009 . . . . . 6 (𝑦 = 𝑎 → (𝐶𝑦𝐶𝑎))
2 fveq2 6905 . . . . . . 7 (𝑦 = 𝑎 → (𝐺𝑦) = (𝐺𝑎))
32sseq2d 4015 . . . . . 6 (𝑦 = 𝑎 → ((𝐺𝐶) ⊆ (𝐺𝑦) ↔ (𝐺𝐶) ⊆ (𝐺𝑎)))
41, 3imbi12d 344 . . . . 5 (𝑦 = 𝑎 → ((𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)) ↔ (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))))
54imbi2d 340 . . . 4 (𝑦 = 𝑎 → (((𝜑𝐶 ∈ On) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦))) ↔ ((𝜑𝐶 ∈ On) → (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)))))
6 sseq2 4009 . . . . . 6 (𝑦 = 𝐷 → (𝐶𝑦𝐶𝐷))
7 fveq2 6905 . . . . . . 7 (𝑦 = 𝐷 → (𝐺𝑦) = (𝐺𝐷))
87sseq2d 4015 . . . . . 6 (𝑦 = 𝐷 → ((𝐺𝐶) ⊆ (𝐺𝑦) ↔ (𝐺𝐶) ⊆ (𝐺𝐷)))
96, 8imbi12d 344 . . . . 5 (𝑦 = 𝐷 → ((𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)) ↔ (𝐶𝐷 → (𝐺𝐶) ⊆ (𝐺𝐷))))
109imbi2d 340 . . . 4 (𝑦 = 𝐷 → (((𝜑𝐶 ∈ On) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦))) ↔ ((𝜑𝐶 ∈ On) → (𝐶𝐷 → (𝐺𝐶) ⊆ (𝐺𝐷)))))
11 r19.21v 3179 . . . . 5 (∀𝑎𝑦 ((𝜑𝐶 ∈ On) → (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) ↔ ((𝜑𝐶 ∈ On) → ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))))
12 onsseleq 6424 . . . . . . . . . 10 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶𝑦 ↔ (𝐶𝑦𝐶 = 𝑦)))
1312ad4ant23 753 . . . . . . . . 9 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → (𝐶𝑦 ↔ (𝐶𝑦𝐶 = 𝑦)))
14 sseq2 4009 . . . . . . . . . . . . 13 (if(𝑦 = ∅, 𝐵, (𝐺𝑦)) = if(𝑦 = 𝑦, if(𝑦 = ∅, 𝐵, (𝐺𝑦)), ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅))) → ((𝐺𝐶) ⊆ if(𝑦 = ∅, 𝐵, (𝐺𝑦)) ↔ (𝐺𝐶) ⊆ if(𝑦 = 𝑦, if(𝑦 = ∅, 𝐵, (𝐺𝑦)), ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅)))))
15 sseq2 4009 . . . . . . . . . . . . 13 (((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅)) = if(𝑦 = 𝑦, if(𝑦 = ∅, 𝐵, (𝐺𝑦)), ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅))) → ((𝐺𝐶) ⊆ ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅)) ↔ (𝐺𝐶) ⊆ if(𝑦 = 𝑦, if(𝑦 = ∅, 𝐵, (𝐺𝑦)), ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅)))))
16 ttukeylem.4 . . . . . . . . . . . . . . . . . 18 𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))
1716tfr1 8438 . . . . . . . . . . . . . . . . 17 𝐺 Fn On
18 simplr 768 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → 𝑦 ∈ On)
19 onss 7806 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ On → 𝑦 ⊆ On)
2018, 19syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → 𝑦 ⊆ On)
21 simprr 772 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → 𝐶𝑦)
22 fnfvima 7254 . . . . . . . . . . . . . . . . 17 ((𝐺 Fn On ∧ 𝑦 ⊆ On ∧ 𝐶𝑦) → (𝐺𝐶) ∈ (𝐺𝑦))
2317, 20, 21, 22mp3an2i 1467 . . . . . . . . . . . . . . . 16 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝐶) ∈ (𝐺𝑦))
24 elssuni 4936 . . . . . . . . . . . . . . . 16 ((𝐺𝐶) ∈ (𝐺𝑦) → (𝐺𝐶) ⊆ (𝐺𝑦))
2523, 24syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝐶) ⊆ (𝐺𝑦))
26 n0i 4339 . . . . . . . . . . . . . . . 16 (𝐶𝑦 → ¬ 𝑦 = ∅)
27 iffalse 4533 . . . . . . . . . . . . . . . 16 𝑦 = ∅ → if(𝑦 = ∅, 𝐵, (𝐺𝑦)) = (𝐺𝑦))
2821, 26, 273syl 18 . . . . . . . . . . . . . . 15 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → if(𝑦 = ∅, 𝐵, (𝐺𝑦)) = (𝐺𝑦))
2925, 28sseqtrrd 4020 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝐶) ⊆ if(𝑦 = ∅, 𝐵, (𝐺𝑦)))
3029adantr 480 . . . . . . . . . . . . 13 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ 𝑦 = 𝑦) → (𝐺𝐶) ⊆ if(𝑦 = ∅, 𝐵, (𝐺𝑦)))
3121adantr 480 . . . . . . . . . . . . . . . 16 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → 𝐶𝑦)
32 elssuni 4936 . . . . . . . . . . . . . . . 16 (𝐶𝑦𝐶 𝑦)
3331, 32syl 17 . . . . . . . . . . . . . . 15 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → 𝐶 𝑦)
34 sseq2 4009 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑦 → (𝐶𝑎𝐶 𝑦))
35 fveq2 6905 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑦 → (𝐺𝑎) = (𝐺 𝑦))
3635sseq2d 4015 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑦 → ((𝐺𝐶) ⊆ (𝐺𝑎) ↔ (𝐺𝐶) ⊆ (𝐺 𝑦)))
3734, 36imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → ((𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ↔ (𝐶 𝑦 → (𝐺𝐶) ⊆ (𝐺 𝑦))))
38 simplrl 776 . . . . . . . . . . . . . . . 16 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)))
39 vuniex 7760 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
4039sucid 6465 . . . . . . . . . . . . . . . . 17 𝑦 ∈ suc 𝑦
41 eloni 6393 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ On → Ord 𝑦)
42 orduniorsuc 7851 . . . . . . . . . . . . . . . . . . 19 (Ord 𝑦 → (𝑦 = 𝑦𝑦 = suc 𝑦))
4318, 41, 423syl 18 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝑦 = 𝑦𝑦 = suc 𝑦))
4443orcanai 1004 . . . . . . . . . . . . . . . . 17 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → 𝑦 = suc 𝑦)
4540, 44eleqtrrid 2847 . . . . . . . . . . . . . . . 16 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → 𝑦𝑦)
4637, 38, 45rspcdva 3622 . . . . . . . . . . . . . . 15 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → (𝐶 𝑦 → (𝐺𝐶) ⊆ (𝐺 𝑦)))
4733, 46mpd 15 . . . . . . . . . . . . . 14 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → (𝐺𝐶) ⊆ (𝐺 𝑦))
48 ssun1 4177 . . . . . . . . . . . . . 14 (𝐺 𝑦) ⊆ ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅))
4947, 48sstrdi 3995 . . . . . . . . . . . . 13 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → (𝐺𝐶) ⊆ ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅)))
5014, 15, 30, 49ifbothda 4563 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝐶) ⊆ if(𝑦 = 𝑦, if(𝑦 = ∅, 𝐵, (𝐺𝑦)), ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅))))
51 ttukeylem.1 . . . . . . . . . . . . . 14 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
52 ttukeylem.2 . . . . . . . . . . . . . 14 (𝜑𝐵𝐴)
53 ttukeylem.3 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
5451, 52, 53, 16ttukeylem3 10552 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ On) → (𝐺𝑦) = if(𝑦 = 𝑦, if(𝑦 = ∅, 𝐵, (𝐺𝑦)), ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅))))
5554ad4ant13 751 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝑦) = if(𝑦 = 𝑦, if(𝑦 = ∅, 𝐵, (𝐺𝑦)), ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅))))
5650, 55sseqtrrd 4020 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝐶) ⊆ (𝐺𝑦))
5756expr 456 . . . . . . . . . 10 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))
58 fveq2 6905 . . . . . . . . . . . 12 (𝐶 = 𝑦 → (𝐺𝐶) = (𝐺𝑦))
59 eqimss 4041 . . . . . . . . . . . 12 ((𝐺𝐶) = (𝐺𝑦) → (𝐺𝐶) ⊆ (𝐺𝑦))
6058, 59syl 17 . . . . . . . . . . 11 (𝐶 = 𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦))
6160a1i 11 . . . . . . . . . 10 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → (𝐶 = 𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))
6257, 61jaod 859 . . . . . . . . 9 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → ((𝐶𝑦𝐶 = 𝑦) → (𝐺𝐶) ⊆ (𝐺𝑦)))
6313, 62sylbid 240 . . . . . . . 8 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))
6463ex 412 . . . . . . 7 (((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) → (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦))))
6564expcom 413 . . . . . 6 (𝑦 ∈ On → ((𝜑𝐶 ∈ On) → (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))))
6665a2d 29 . . . . 5 (𝑦 ∈ On → (((𝜑𝐶 ∈ On) → ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → ((𝜑𝐶 ∈ On) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))))
6711, 66biimtrid 242 . . . 4 (𝑦 ∈ On → (∀𝑎𝑦 ((𝜑𝐶 ∈ On) → (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → ((𝜑𝐶 ∈ On) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))))
685, 10, 67tfis3 7880 . . 3 (𝐷 ∈ On → ((𝜑𝐶 ∈ On) → (𝐶𝐷 → (𝐺𝐶) ⊆ (𝐺𝐷))))
6968expdcom 414 . 2 (𝜑 → (𝐶 ∈ On → (𝐷 ∈ On → (𝐶𝐷 → (𝐺𝐶) ⊆ (𝐺𝐷)))))
70693imp2 1349 1 ((𝜑 ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝐶𝐷)) → (𝐺𝐶) ⊆ (𝐺𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086  wal 1537   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  cdif 3947  cun 3948  cin 3949  wss 3950  c0 4332  ifcif 4524  𝒫 cpw 4599  {csn 4625   cuni 4906  cmpt 5224  dom cdm 5684  ran crn 5685  cima 5687  Ord word 6382  Oncon0 6383  suc csuc 6385   Fn wfn 6555  1-1-ontowf1o 6559  cfv 6560  recscrecs 8411  Fincfn 8986  cardccrd 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412
This theorem is referenced by:  ttukeylem6  10555  ttukeylem7  10556
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