Theorem ttukeylem5 9784

Description: Lemma for ttukey 9789. The 𝐺 function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.)

Hypotheses

Ref	Expression
ttukeylem.1	⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
ttukeylem.2	⊢ (𝜑 → 𝐵 ∈ 𝐴)
ttukeylem.3	⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (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 ∧ 𝐶 ⊆ 𝐷)) → (𝐺‘𝐶) ⊆ (𝐺‘𝐷))

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.

Step	Hyp	Ref	Expression
1		sseq2 3916	. . . . . 6 ⊢ (𝑦 = 𝑎 → (𝐶 ⊆ 𝑦 ↔ 𝐶 ⊆ 𝑎))
2		fveq2 6541	. . . . . . 7 ⊢ (𝑦 = 𝑎 → (𝐺‘𝑦) = (𝐺‘𝑎))
3	2	sseq2d 3922	. . . . . 6 ⊢ (𝑦 = 𝑎 → ((𝐺‘𝐶) ⊆ (𝐺‘𝑦) ↔ (𝐺‘𝐶) ⊆ (𝐺‘𝑎)))
4	1, 3	imbi12d 346	. . . . 5 ⊢ (𝑦 = 𝑎 → ((𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦)) ↔ (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎))))
5	4	imbi2d 342	. . . 4 ⊢ (𝑦 = 𝑎 → (((𝜑 ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦))) ↔ ((𝜑 ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)))))
6		sseq2 3916	. . . . . 6 ⊢ (𝑦 = 𝐷 → (𝐶 ⊆ 𝑦 ↔ 𝐶 ⊆ 𝐷))
7		fveq2 6541	. . . . . . 7 ⊢ (𝑦 = 𝐷 → (𝐺‘𝑦) = (𝐺‘𝐷))
8	7	sseq2d 3922	. . . . . 6 ⊢ (𝑦 = 𝐷 → ((𝐺‘𝐶) ⊆ (𝐺‘𝑦) ↔ (𝐺‘𝐶) ⊆ (𝐺‘𝐷)))
9	6, 8	imbi12d 346	. . . . 5 ⊢ (𝑦 = 𝐷 → ((𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦)) ↔ (𝐶 ⊆ 𝐷 → (𝐺‘𝐶) ⊆ (𝐺‘𝐷))))
10	9	imbi2d 342	. . . 4 ⊢ (𝑦 = 𝐷 → (((𝜑 ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦))) ↔ ((𝜑 ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝐷 → (𝐺‘𝐶) ⊆ (𝐺‘𝐷)))))
11		r19.21v 3141	. . . . 5 ⊢ (∀𝑎 ∈ 𝑦 ((𝜑 ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎))) ↔ ((𝜑 ∧ 𝐶 ∈ On) → ∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎))))
12		onsseleq 6110	. . . . . . . . . 10 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ⊆ 𝑦 ↔ (𝐶 ∈ 𝑦 ∨ 𝐶 = 𝑦)))
13	12	ad4ant23 749	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎))) → (𝐶 ⊆ 𝑦 ↔ (𝐶 ∈ 𝑦 ∨ 𝐶 = 𝑦)))
14		sseq2 3916	. . . . . . . . . . . . 13 ⊢ (if(𝑦 = ∅, 𝐵, ∪ (𝐺 “ 𝑦)) = if(𝑦 = ∪ 𝑦, if(𝑦 = ∅, 𝐵, ∪ (𝐺 “ 𝑦)), ((𝐺‘∪ 𝑦) ∪ if(((𝐺‘∪ 𝑦) ∪ {(𝐹‘∪ 𝑦)}) ∈ 𝐴, {(𝐹‘∪ 𝑦)}, ∅))) → ((𝐺‘𝐶) ⊆ if(𝑦 = ∅, 𝐵, ∪ (𝐺 “ 𝑦)) ↔ (𝐺‘𝐶) ⊆ if(𝑦 = ∪ 𝑦, if(𝑦 = ∅, 𝐵, ∪ (𝐺 “ 𝑦)), ((𝐺‘∪ 𝑦) ∪ if(((𝐺‘∪ 𝑦) ∪ {(𝐹‘∪ 𝑦)}) ∈ 𝐴, {(𝐹‘∪ 𝑦)}, ∅)))))
15		sseq2 3916	. . . . . . . . . . . . 13 ⊢ (((𝐺‘∪ 𝑦) ∪ if(((𝐺‘∪ 𝑦) ∪ {(𝐹‘∪ 𝑦)}) ∈ 𝐴, {(𝐹‘∪ 𝑦)}, ∅)) = if(𝑦 = ∪ 𝑦, if(𝑦 = ∅, 𝐵, ∪ (𝐺 “ 𝑦)), ((𝐺‘∪ 𝑦) ∪ if(((𝐺‘∪ 𝑦) ∪ {(𝐹‘∪ 𝑦)}) ∈ 𝐴, {(𝐹‘∪ 𝑦)}, ∅))) → ((𝐺‘𝐶) ⊆ ((𝐺‘∪ 𝑦) ∪ if(((𝐺‘∪ 𝑦) ∪ {(𝐹‘∪ 𝑦)}) ∈ 𝐴, {(𝐹‘∪ 𝑦)}, ∅)) ↔ (𝐺‘𝐶) ⊆ if(𝑦 = ∪ 𝑦, if(𝑦 = ∅, 𝐵, ∪ (𝐺 “ 𝑦)), ((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)}, ∅)))))
17	16	tfr1 7888	. . . . . . . . . . . . . . . . 17 ⊢ 𝐺 Fn On
18		simplr 765	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → 𝑦 ∈ On)
19		onss 7364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ On → 𝑦 ⊆ On)
20	18, 19	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → 𝑦 ⊆ On)
21		simprr 769	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → 𝐶 ∈ 𝑦)
22		fnfvima 6863	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 Fn On ∧ 𝑦 ⊆ On ∧ 𝐶 ∈ 𝑦) → (𝐺‘𝐶) ∈ (𝐺 “ 𝑦))
23	17, 20, 21, 22	mp3an2i 1458	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → (𝐺‘𝐶) ∈ (𝐺 “ 𝑦))
24		elssuni 4776	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝐶) ∈ (𝐺 “ 𝑦) → (𝐺‘𝐶) ⊆ ∪ (𝐺 “ 𝑦))
25	23, 24	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → (𝐺‘𝐶) ⊆ ∪ (𝐺 “ 𝑦))
26		n0i 4221	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ 𝑦 → ¬ 𝑦 = ∅)
27		iffalse 4392	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑦 = ∅ → if(𝑦 = ∅, 𝐵, ∪ (𝐺 “ 𝑦)) = ∪ (𝐺 “ 𝑦))
28	21, 26, 27	3syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → if(𝑦 = ∅, 𝐵, ∪ (𝐺 “ 𝑦)) = ∪ (𝐺 “ 𝑦))
29	25, 28	sseqtr4d 3931	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → (𝐺‘𝐶) ⊆ if(𝑦 = ∅, 𝐵, ∪ (𝐺 “ 𝑦)))
30	29	adantr 481	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) ∧ 𝑦 = ∪ 𝑦) → (𝐺‘𝐶) ⊆ if(𝑦 = ∅, 𝐵, ∪ (𝐺 “ 𝑦)))
31	21	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) ∧ ¬ 𝑦 = ∪ 𝑦) → 𝐶 ∈ 𝑦)
32		elssuni 4776	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ 𝑦 → 𝐶 ⊆ ∪ 𝑦)
33	31, 32	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) ∧ ¬ 𝑦 = ∪ 𝑦) → 𝐶 ⊆ ∪ 𝑦)
34		sseq2 3916	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = ∪ 𝑦 → (𝐶 ⊆ 𝑎 ↔ 𝐶 ⊆ ∪ 𝑦))
35		fveq2 6541	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = ∪ 𝑦 → (𝐺‘𝑎) = (𝐺‘∪ 𝑦))
36	35	sseq2d 3922	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = ∪ 𝑦 → ((𝐺‘𝐶) ⊆ (𝐺‘𝑎) ↔ (𝐺‘𝐶) ⊆ (𝐺‘∪ 𝑦)))
37	34, 36	imbi12d 346	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = ∪ 𝑦 → ((𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ↔ (𝐶 ⊆ ∪ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘∪ 𝑦))))
38		simplrl 773	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) ∧ ¬ 𝑦 = ∪ 𝑦) → ∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)))
39		vuniex 7327	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑦 ∈ V
40	39	sucid 6148	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑦 ∈ suc ∪ 𝑦
41		eloni 6079	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ On → Ord 𝑦)
42		orduniorsuc 7404	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord 𝑦 → (𝑦 = ∪ 𝑦 ∨ 𝑦 = suc ∪ 𝑦))
43	18, 41, 42	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → (𝑦 = ∪ 𝑦 ∨ 𝑦 = suc ∪ 𝑦))
44	43	orcanai 997	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) ∧ ¬ 𝑦 = ∪ 𝑦) → 𝑦 = suc ∪ 𝑦)
45	40, 44	syl5eleqr 2889	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) ∧ ¬ 𝑦 = ∪ 𝑦) → ∪ 𝑦 ∈ 𝑦)
46	37, 38, 45	rspcdva 3563	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) ∧ ¬ 𝑦 = ∪ 𝑦) → (𝐶 ⊆ ∪ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘∪ 𝑦)))
47	33, 46	mpd 15	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) ∧ ¬ 𝑦 = ∪ 𝑦) → (𝐺‘𝐶) ⊆ (𝐺‘∪ 𝑦))
48		ssun1 4071	. . . . . . . . . . . . . 14 ⊢ (𝐺‘∪ 𝑦) ⊆ ((𝐺‘∪ 𝑦) ∪ if(((𝐺‘∪ 𝑦) ∪ {(𝐹‘∪ 𝑦)}) ∈ 𝐴, {(𝐹‘∪ 𝑦)}, ∅))
49	47, 48	syl6ss 3903	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) ∧ ¬ 𝑦 = ∪ 𝑦) → (𝐺‘𝐶) ⊆ ((𝐺‘∪ 𝑦) ∪ if(((𝐺‘∪ 𝑦) ∪ {(𝐹‘∪ 𝑦)}) ∈ 𝐴, {(𝐹‘∪ 𝑦)}, ∅)))
50	14, 15, 30, 49	ifbothda 4420	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → (𝐺‘𝐶) ⊆ if(𝑦 = ∪ 𝑦, if(𝑦 = ∅, 𝐵, ∪ (𝐺 “ 𝑦)), ((𝐺‘∪ 𝑦) ∪ if(((𝐺‘∪ 𝑦) ∪ {(𝐹‘∪ 𝑦)}) ∈ 𝐴, {(𝐹‘∪ 𝑦)}, ∅))))
51		ttukeylem.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
52		ttukeylem.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
53		ttukeylem.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
54	51, 52, 53, 16	ttukeylem3 9782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ On) → (𝐺‘𝑦) = if(𝑦 = ∪ 𝑦, if(𝑦 = ∅, 𝐵, ∪ (𝐺 “ 𝑦)), ((𝐺‘∪ 𝑦) ∪ if(((𝐺‘∪ 𝑦) ∪ {(𝐹‘∪ 𝑦)}) ∈ 𝐴, {(𝐹‘∪ 𝑦)}, ∅))))
55	54	ad4ant13 747	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → (𝐺‘𝑦) = if(𝑦 = ∪ 𝑦, if(𝑦 = ∅, 𝐵, ∪ (𝐺 “ 𝑦)), ((𝐺‘∪ 𝑦) ∪ if(((𝐺‘∪ 𝑦) ∪ {(𝐹‘∪ 𝑦)}) ∈ 𝐴, {(𝐹‘∪ 𝑦)}, ∅))))
56	50, 55	sseqtr4d 3931	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → (𝐺‘𝐶) ⊆ (𝐺‘𝑦))
57	56	expr 457	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎))) → (𝐶 ∈ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦)))
58		fveq2 6541	. . . . . . . . . . . 12 ⊢ (𝐶 = 𝑦 → (𝐺‘𝐶) = (𝐺‘𝑦))
59		eqimss 3946	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝐶) = (𝐺‘𝑦) → (𝐺‘𝐶) ⊆ (𝐺‘𝑦))
60	58, 59	syl 17	. . . . . . . . . . 11 ⊢ (𝐶 = 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦))
61	60	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎))) → (𝐶 = 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦)))
62	57, 61	jaod 854	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎))) → ((𝐶 ∈ 𝑦 ∨ 𝐶 = 𝑦) → (𝐺‘𝐶) ⊆ (𝐺‘𝑦)))
63	13, 62	sylbid 241	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎))) → (𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦)))
64	63	ex 413	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) → (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) → (𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦))))
65	64	expcom 414	. . . . . 6 ⊢ (𝑦 ∈ On → ((𝜑 ∧ 𝐶 ∈ On) → (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) → (𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦)))))
66	65	a2d 29	. . . . 5 ⊢ (𝑦 ∈ On → (((𝜑 ∧ 𝐶 ∈ On) → ∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎))) → ((𝜑 ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦)))))
67	11, 66	syl5bi 243	. . . 4 ⊢ (𝑦 ∈ On → (∀𝑎 ∈ 𝑦 ((𝜑 ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎))) → ((𝜑 ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦)))))
68	5, 10, 67	tfis3 7431	. . 3 ⊢ (𝐷 ∈ On → ((𝜑 ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝐷 → (𝐺‘𝐶) ⊆ (𝐺‘𝐷))))
69	68	expdcom 415	. 2 ⊢ (𝜑 → (𝐶 ∈ On → (𝐷 ∈ On → (𝐶 ⊆ 𝐷 → (𝐺‘𝐶) ⊆ (𝐺‘𝐷)))))
70	69	3imp2 1342	1 ⊢ ((𝜑 ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝐶 ⊆ 𝐷)) → (𝐺‘𝐶) ⊆ (𝐺‘𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   ∧ w3a 1080  ∀wal 1520   = wceq 1522   ∈ wcel 2080  ∀wral 3104  Vcvv 3436   ∖ cdif 3858   ∪ cun 3859   ∩ cin 3860   ⊆ wss 3861  ∅c0 4213  ifcif 4383  𝒫 cpw 4455  {csn 4474  ∪ cuni 4747   ↦ cmpt 5043  dom cdm 5446  ran crn 5447   “ cima 5449  Ord word 6068  Oncon0 6069  suc csuc 6071   Fn wfn 6223  –1-1-onto→wf1o 6227  ‘cfv 6228  recscrecs 7862  Fincfn 8360  cardccrd 9213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-rep 5084  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-reu 3111  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-pred 6026  df-ord 6072  df-on 6073  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-wrecs 7801  df-recs 7863

This theorem is referenced by:  ttukeylem6  9785  ttukeylem7  9786