Theorem ttukeylem5 10519

Description: Lemma for ttukey 10524. 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 3960	. . . . . 6 ⊢ (𝑦 = 𝑎 → (𝐶 ⊆ 𝑦 ↔ 𝐶 ⊆ 𝑎))
2		fveq2 6882	. . . . . . 7 ⊢ (𝑦 = 𝑎 → (𝐺‘𝑦) = (𝐺‘𝑎))
3	2	sseq2d 3966	. . . . . 6 ⊢ (𝑦 = 𝑎 → ((𝐺‘𝐶) ⊆ (𝐺‘𝑦) ↔ (𝐺‘𝐶) ⊆ (𝐺‘𝑎)))
4	1, 3	imbi12d 347	. . . . 5 ⊢ (𝑦 = 𝑎 → ((𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦)) ↔ (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎))))
5	4	imbi2d 343	. . . 4 ⊢ (𝑦 = 𝑎 → (((𝜑 ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦))) ↔ ((𝜑 ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)))))
6		sseq2 3960	. . . . . 6 ⊢ (𝑦 = 𝐷 → (𝐶 ⊆ 𝑦 ↔ 𝐶 ⊆ 𝐷))
7		fveq2 6882	. . . . . . 7 ⊢ (𝑦 = 𝐷 → (𝐺‘𝑦) = (𝐺‘𝐷))
8	7	sseq2d 3966	. . . . . 6 ⊢ (𝑦 = 𝐷 → ((𝐺‘𝐶) ⊆ (𝐺‘𝑦) ↔ (𝐺‘𝐶) ⊆ (𝐺‘𝐷)))
9	6, 8	imbi12d 347	. . . . 5 ⊢ (𝑦 = 𝐷 → ((𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦)) ↔ (𝐶 ⊆ 𝐷 → (𝐺‘𝐶) ⊆ (𝐺‘𝐷))))
10	9	imbi2d 343	. . . 4 ⊢ (𝑦 = 𝐷 → (((𝜑 ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦))) ↔ ((𝜑 ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝐷 → (𝐺‘𝐶) ⊆ (𝐺‘𝐷)))))
11		r19.21v 3189	. . . . 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 6403	. . . . . . . . . 10 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ⊆ 𝑦 ↔ (𝐶 ∈ 𝑦 ∨ 𝐶 = 𝑦)))
13	12	ad4ant23 766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎))) → (𝐶 ⊆ 𝑦 ↔ (𝐶 ∈ 𝑦 ∨ 𝐶 = 𝑦)))
14		sseq2 3960	. . . . . . . . . . . . 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 3960	. . . . . . . . . . . . 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 8390	. . . . . . . . . . . . . . . . 17 ⊢ 𝐺 Fn On
18		simplr 781	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → 𝑦 ∈ On)
19		onss 7788	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ On → 𝑦 ⊆ On)
20	18, 19	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → 𝑦 ⊆ On)
21		simprr 785	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → 𝐶 ∈ 𝑦)
22		fnfvima 7236	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 Fn On ∧ 𝑦 ⊆ On ∧ 𝐶 ∈ 𝑦) → (𝐺‘𝐶) ∈ (𝐺 “ 𝑦))
23	17, 20, 21, 22	mp3an2i 1495	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → (𝐺‘𝐶) ∈ (𝐺 “ 𝑦))
24		elssuni 4902	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝐶) ∈ (𝐺 “ 𝑦) → (𝐺‘𝐶) ⊆ ∪ (𝐺 “ 𝑦))
25	23, 24	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → (𝐺‘𝐶) ⊆ ∪ (𝐺 “ 𝑦))
26		n0i 4289	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ 𝑦 → ¬ 𝑦 = ∅)
27		iffalse 4494	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑦 = ∅ → if(𝑦 = ∅, 𝐵, ∪ (𝐺 “ 𝑦)) = ∪ (𝐺 “ 𝑦))
28	21, 26, 27	3syl 19	. . . . . . . . . . . . . . 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	sseqtrrd 3971	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) ∧ ¬ 𝑦 = ∪ 𝑦) → 𝐶 ∈ 𝑦)
32		elssuni 4902	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ 𝑦 → 𝐶 ⊆ ∪ 𝑦)
33	31, 32	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) ∧ ¬ 𝑦 = ∪ 𝑦) → 𝐶 ⊆ ∪ 𝑦)
34		sseq2 3960	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = ∪ 𝑦 → (𝐶 ⊆ 𝑎 ↔ 𝐶 ⊆ ∪ 𝑦))
35		fveq2 6882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = ∪ 𝑦 → (𝐺‘𝑎) = (𝐺‘∪ 𝑦))
36	35	sseq2d 3966	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = ∪ 𝑦 → ((𝐺‘𝐶) ⊆ (𝐺‘𝑎) ↔ (𝐺‘𝐶) ⊆ (𝐺‘∪ 𝑦)))
37	34, 36	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = ∪ 𝑦 → ((𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ↔ (𝐶 ⊆ ∪ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘∪ 𝑦))))
38		simplrl 789	. . . . . . . . . . . . . . . 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 7745	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑦 ∈ V
40	39	sucid 6446	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑦 ∈ suc ∪ 𝑦
41		eloni 6371	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ On → Ord 𝑦)
42		orduniorsuc 7830	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord 𝑦 → (𝑦 = ∪ 𝑦 ∨ 𝑦 = suc ∪ 𝑦))
43	18, 41, 42	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → (𝑦 = ∪ 𝑦 ∨ 𝑦 = suc ∪ 𝑦))
44	43	orcanai 1018	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) ∧ ¬ 𝑦 = ∪ 𝑦) → 𝑦 = suc ∪ 𝑦)
45	40, 44	eleqtrrid 2869	. . . . . . . . . . . . . . . 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 3580	. . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) ∧ ¬ 𝑦 = ∪ 𝑦) → (𝐺‘𝐶) ⊆ (𝐺‘∪ 𝑦))
48		ssun1 4127	. . . . . . . . . . . . . 14 ⊢ (𝐺‘∪ 𝑦) ⊆ ((𝐺‘∪ 𝑦) ∪ if(((𝐺‘∪ 𝑦) ∪ {(𝐹‘∪ 𝑦)}) ∈ 𝐴, {(𝐹‘∪ 𝑦)}, ∅))
49	47, 48	sstrdi 3946	. . . . . . . . . . . . 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 4524	. . . . . . . . . . . 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 10517	. . . . . . . . . . . . 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 764	. . . . . . . . . . . 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	sseqtrrd 3971	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) ∧ 𝐶 ∈ 𝑦)) → (𝐺‘𝐶) ⊆ (𝐺‘𝑦))
57	56	expr 462	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎))) → (𝐶 ∈ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦)))
58		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝐶 = 𝑦 → (𝐺‘𝐶) = (𝐺‘𝑦))
59		eqimss 3992	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝐶) = (𝐺‘𝑦) → (𝐺‘𝐶) ⊆ (𝐺‘𝑦))
60	58, 59	syl 18	. . . . . . . . . . 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 873	. . . . . . . . 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 243	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎))) → (𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦)))
64	63	ex 418	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ On) ∧ 𝑦 ∈ On) → (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) → (𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦))))
65	64	expcom 419	. . . . . 6 ⊢ (𝑦 ∈ On → ((𝜑 ∧ 𝐶 ∈ On) → (∀𝑎 ∈ 𝑦 (𝐶 ⊆ 𝑎 → (𝐺‘𝐶) ⊆ (𝐺‘𝑎)) → (𝐶 ⊆ 𝑦 → (𝐺‘𝐶) ⊆ (𝐺‘𝑦)))))
66	65	a2d 30	. . . . 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	biimtrid 245	. . . 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 7858	. . 3 ⊢ (𝐷 ∈ On → ((𝜑 ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝐷 → (𝐺‘𝐶) ⊆ (𝐺‘𝐷))))
69	68	expdcom 420	. 2 ⊢ (𝜑 → (𝐶 ∈ On → (𝐷 ∈ On → (𝐶 ⊆ 𝐷 → (𝐺‘𝐶) ⊆ (𝐺‘𝐷)))))
70	69	3imp2 1368	1 ⊢ ((𝜑 ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝐶 ⊆ 𝐷)) → (𝐺‘𝐶) ⊆ (𝐺‘𝐷))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  ifcif 4485  𝒫 cpw 4560  {csn 4587  ∪ cuni 4870   ↦ cmpt 5190  dom cdm 5659  ran crn 5660   “ cima 5662  Ord word 6360  Oncon0 6361  suc csuc 6363   Fn wfn 6532  –1-1-onto→wf1o 6536  ‘cfv 6537  recscrecs 8363  Fincfn 8956  cardccrd 9944

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364

This theorem is used by:  ttukeylem6  10520  ttukeylem7  10521