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Theorem ttukeylem5 9320
Description: Lemma for ttukey 9325. 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 3619 . . . . . 6 (𝑦 = 𝑎 → (𝐶𝑦𝐶𝑎))
2 fveq2 6178 . . . . . . 7 (𝑦 = 𝑎 → (𝐺𝑦) = (𝐺𝑎))
32sseq2d 3625 . . . . . 6 (𝑦 = 𝑎 → ((𝐺𝐶) ⊆ (𝐺𝑦) ↔ (𝐺𝐶) ⊆ (𝐺𝑎)))
41, 3imbi12d 334 . . . . 5 (𝑦 = 𝑎 → ((𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)) ↔ (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))))
54imbi2d 330 . . . 4 (𝑦 = 𝑎 → (((𝜑𝐶 ∈ On) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦))) ↔ ((𝜑𝐶 ∈ On) → (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)))))
6 sseq2 3619 . . . . . 6 (𝑦 = 𝐷 → (𝐶𝑦𝐶𝐷))
7 fveq2 6178 . . . . . . 7 (𝑦 = 𝐷 → (𝐺𝑦) = (𝐺𝐷))
87sseq2d 3625 . . . . . 6 (𝑦 = 𝐷 → ((𝐺𝐶) ⊆ (𝐺𝑦) ↔ (𝐺𝐶) ⊆ (𝐺𝐷)))
96, 8imbi12d 334 . . . . 5 (𝑦 = 𝐷 → ((𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)) ↔ (𝐶𝐷 → (𝐺𝐶) ⊆ (𝐺𝐷))))
109imbi2d 330 . . . 4 (𝑦 = 𝐷 → (((𝜑𝐶 ∈ On) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦))) ↔ ((𝜑𝐶 ∈ On) → (𝐶𝐷 → (𝐺𝐶) ⊆ (𝐺𝐷)))))
11 r19.21v 2957 . . . . 5 (∀𝑎𝑦 ((𝜑𝐶 ∈ On) → (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) ↔ ((𝜑𝐶 ∈ On) → ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))))
12 simpllr 798 . . . . . . . . . 10 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → 𝐶 ∈ On)
13 simplr 791 . . . . . . . . . 10 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → 𝑦 ∈ On)
14 onsseleq 5753 . . . . . . . . . 10 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶𝑦 ↔ (𝐶𝑦𝐶 = 𝑦)))
1512, 13, 14syl2anc 692 . . . . . . . . 9 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → (𝐶𝑦 ↔ (𝐶𝑦𝐶 = 𝑦)))
16 sseq2 3619 . . . . . . . . . . . . 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)}, ∅)))))
17 sseq2 3619 . . . . . . . . . . . . 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)}, ∅)))))
18 ttukeylem.4 . . . . . . . . . . . . . . . . . . 19 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)}, ∅)))))
1918tfr1 7478 . . . . . . . . . . . . . . . . . 18 𝐺 Fn On
2019a1i 11 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → 𝐺 Fn On)
21 simplr 791 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → 𝑦 ∈ On)
22 onss 6975 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ On → 𝑦 ⊆ On)
2321, 22syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → 𝑦 ⊆ On)
24 simprr 795 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → 𝐶𝑦)
25 fnfvima 6481 . . . . . . . . . . . . . . . . 17 ((𝐺 Fn On ∧ 𝑦 ⊆ On ∧ 𝐶𝑦) → (𝐺𝐶) ∈ (𝐺𝑦))
2620, 23, 24, 25syl3anc 1324 . . . . . . . . . . . . . . . 16 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝐶) ∈ (𝐺𝑦))
27 elssuni 4458 . . . . . . . . . . . . . . . 16 ((𝐺𝐶) ∈ (𝐺𝑦) → (𝐺𝐶) ⊆ (𝐺𝑦))
2826, 27syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝐶) ⊆ (𝐺𝑦))
29 n0i 3912 . . . . . . . . . . . . . . . 16 (𝐶𝑦 → ¬ 𝑦 = ∅)
30 iffalse 4086 . . . . . . . . . . . . . . . 16 𝑦 = ∅ → if(𝑦 = ∅, 𝐵, (𝐺𝑦)) = (𝐺𝑦))
3124, 29, 303syl 18 . . . . . . . . . . . . . . 15 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → if(𝑦 = ∅, 𝐵, (𝐺𝑦)) = (𝐺𝑦))
3228, 31sseqtr4d 3634 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝐶) ⊆ if(𝑦 = ∅, 𝐵, (𝐺𝑦)))
3332adantr 481 . . . . . . . . . . . . 13 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ 𝑦 = 𝑦) → (𝐺𝐶) ⊆ if(𝑦 = ∅, 𝐵, (𝐺𝑦)))
34 vuniex 6939 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
3534sucid 5792 . . . . . . . . . . . . . . . 16 𝑦 ∈ suc 𝑦
36 eloni 5721 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ On → Ord 𝑦)
37 orduniorsuc 7015 . . . . . . . . . . . . . . . . . 18 (Ord 𝑦 → (𝑦 = 𝑦𝑦 = suc 𝑦))
3821, 36, 373syl 18 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝑦 = 𝑦𝑦 = suc 𝑦))
3938orcanai 951 . . . . . . . . . . . . . . . 16 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → 𝑦 = suc 𝑦)
4035, 39syl5eleqr 2706 . . . . . . . . . . . . . . 15 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → 𝑦𝑦)
41 simplrl 799 . . . . . . . . . . . . . . 15 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)))
4224adantr 481 . . . . . . . . . . . . . . . 16 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → 𝐶𝑦)
43 elssuni 4458 . . . . . . . . . . . . . . . 16 (𝐶𝑦𝐶 𝑦)
4442, 43syl 17 . . . . . . . . . . . . . . 15 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → 𝐶 𝑦)
45 sseq2 3619 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑦 → (𝐶𝑎𝐶 𝑦))
46 fveq2 6178 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑦 → (𝐺𝑎) = (𝐺 𝑦))
4746sseq2d 3625 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑦 → ((𝐺𝐶) ⊆ (𝐺𝑎) ↔ (𝐺𝐶) ⊆ (𝐺 𝑦)))
4845, 47imbi12d 334 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → ((𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ↔ (𝐶 𝑦 → (𝐺𝐶) ⊆ (𝐺 𝑦))))
4948rspcv 3300 . . . . . . . . . . . . . . 15 ( 𝑦𝑦 → (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) → (𝐶 𝑦 → (𝐺𝐶) ⊆ (𝐺 𝑦))))
5040, 41, 44, 49syl3c 66 . . . . . . . . . . . . . 14 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → (𝐺𝐶) ⊆ (𝐺 𝑦))
51 ssun1 3768 . . . . . . . . . . . . . 14 (𝐺 𝑦) ⊆ ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅))
5250, 51syl6ss 3607 . . . . . . . . . . . . 13 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → (𝐺𝐶) ⊆ ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅)))
5316, 17, 33, 52ifbothda 4114 . . . . . . . . . . . 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)}, ∅))))
54 simplll 797 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → 𝜑)
55 ttukeylem.1 . . . . . . . . . . . . . 14 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
56 ttukeylem.2 . . . . . . . . . . . . . 14 (𝜑𝐵𝐴)
57 ttukeylem.3 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
5855, 56, 57, 18ttukeylem3 9318 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ On) → (𝐺𝑦) = if(𝑦 = 𝑦, if(𝑦 = ∅, 𝐵, (𝐺𝑦)), ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅))))
5954, 21, 58syl2anc 692 . . . . . . . . . . . 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)}, ∅))))
6053, 59sseqtr4d 3634 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝐶) ⊆ (𝐺𝑦))
6160expr 642 . . . . . . . . . 10 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))
62 fveq2 6178 . . . . . . . . . . . 12 (𝐶 = 𝑦 → (𝐺𝐶) = (𝐺𝑦))
63 eqimss 3649 . . . . . . . . . . . 12 ((𝐺𝐶) = (𝐺𝑦) → (𝐺𝐶) ⊆ (𝐺𝑦))
6462, 63syl 17 . . . . . . . . . . 11 (𝐶 = 𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦))
6564a1i 11 . . . . . . . . . 10 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → (𝐶 = 𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))
6661, 65jaod 395 . . . . . . . . 9 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → ((𝐶𝑦𝐶 = 𝑦) → (𝐺𝐶) ⊆ (𝐺𝑦)))
6715, 66sylbid 230 . . . . . . . 8 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))
6867ex 450 . . . . . . 7 (((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) → (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦))))
6968expcom 451 . . . . . 6 (𝑦 ∈ On → ((𝜑𝐶 ∈ On) → (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))))
7069a2d 29 . . . . 5 (𝑦 ∈ On → (((𝜑𝐶 ∈ On) → ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → ((𝜑𝐶 ∈ On) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))))
7111, 70syl5bi 232 . . . 4 (𝑦 ∈ On → (∀𝑎𝑦 ((𝜑𝐶 ∈ On) → (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → ((𝜑𝐶 ∈ On) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))))
725, 10, 71tfis3 7042 . . 3 (𝐷 ∈ On → ((𝜑𝐶 ∈ On) → (𝐶𝐷 → (𝐺𝐶) ⊆ (𝐺𝐷))))
7372expdcom 455 . 2 (𝜑 → (𝐶 ∈ On → (𝐷 ∈ On → (𝐶𝐷 → (𝐺𝐶) ⊆ (𝐺𝐷)))))
74733imp2 1280 1 ((𝜑 ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝐶𝐷)) → (𝐺𝐶) ⊆ (𝐺𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036  wal 1479   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  cdif 3564  cun 3565  cin 3566  wss 3567  c0 3907  ifcif 4077  𝒫 cpw 4149  {csn 4168   cuni 4427  cmpt 4720  dom cdm 5104  ran crn 5105  cima 5107  Ord word 5710  Oncon0 5711  suc csuc 5713   Fn wfn 5871  1-1-ontowf1o 5875  cfv 5876  recscrecs 7452  Fincfn 7940  cardccrd 8746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-wrecs 7392  df-recs 7453
This theorem is referenced by:  ttukeylem6  9321  ttukeylem7  9322
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