Theorem isotone2 45048

Description: Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021.)

Assertion

Ref	Expression
isotone2	⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)))

Distinct variable groups:   𝐴,𝑎,𝑏   𝐹,𝑎,𝑏

Proof of Theorem isotone2

Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3956	. . . 4 ⊢ (𝑎 = 𝑐 → (𝑎 ⊆ 𝑏 ↔ 𝑐 ⊆ 𝑏))
2		fveq2 6885	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝐹‘𝑎) = (𝐹‘𝑐))
3	2	sseq1d 3962	. . . 4 ⊢ (𝑎 = 𝑐 → ((𝐹‘𝑎) ⊆ (𝐹‘𝑏) ↔ (𝐹‘𝑐) ⊆ (𝐹‘𝑏)))
4	1, 3	imbi12d 347	. . 3 ⊢ (𝑎 = 𝑐 → ((𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ (𝑐 ⊆ 𝑏 → (𝐹‘𝑐) ⊆ (𝐹‘𝑏))))
5		sseq2 3957	. . . 4 ⊢ (𝑏 = 𝑑 → (𝑐 ⊆ 𝑏 ↔ 𝑐 ⊆ 𝑑))
6		fveq2 6885	. . . . 5 ⊢ (𝑏 = 𝑑 → (𝐹‘𝑏) = (𝐹‘𝑑))
7	6	sseq2d 3963	. . . 4 ⊢ (𝑏 = 𝑑 → ((𝐹‘𝑐) ⊆ (𝐹‘𝑏) ↔ (𝐹‘𝑐) ⊆ (𝐹‘𝑑)))
8	5, 7	imbi12d 347	. . 3 ⊢ (𝑏 = 𝑑 → ((𝑐 ⊆ 𝑏 → (𝐹‘𝑐) ⊆ (𝐹‘𝑏)) ↔ (𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑))))
9	4, 8	cbvral2vw 3245	. 2 ⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ ∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)))
10		inss1 4182	. . . . . 6 ⊢ (𝑎 ∩ 𝑏) ⊆ 𝑎
11		inss2 4183	. . . . . . . . . 10 ⊢ (𝑎 ∩ 𝑏) ⊆ 𝑏
12		elpwi 4564	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
13	11, 12	sstrid 3942	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝒫 𝐴 → (𝑎 ∩ 𝑏) ⊆ 𝐴)
14		vex 3455	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
15	14	inex2 5278	. . . . . . . . . 10 ⊢ (𝑎 ∩ 𝑏) ∈ V
16	15	elpw 4561	. . . . . . . . 9 ⊢ ((𝑎 ∩ 𝑏) ∈ 𝒫 𝐴 ↔ (𝑎 ∩ 𝑏) ⊆ 𝐴)
17	13, 16	sylibr 237	. . . . . . . 8 ⊢ (𝑏 ∈ 𝒫 𝐴 → (𝑎 ∩ 𝑏) ∈ 𝒫 𝐴)
18	17	ad2antll 742	. . . . . . 7 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝑎 ∩ 𝑏) ∈ 𝒫 𝐴)
19		simprl 783	. . . . . . 7 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → 𝑎 ∈ 𝒫 𝐴)
20		simpl 488	. . . . . . 7 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → ∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)))
21		sseq1 3956	. . . . . . . . 9 ⊢ (𝑐 = (𝑎 ∩ 𝑏) → (𝑐 ⊆ 𝑑 ↔ (𝑎 ∩ 𝑏) ⊆ 𝑑))
22		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑐 = (𝑎 ∩ 𝑏) → (𝐹‘𝑐) = (𝐹‘(𝑎 ∩ 𝑏)))
23	22	sseq1d 3962	. . . . . . . . 9 ⊢ (𝑐 = (𝑎 ∩ 𝑏) → ((𝐹‘𝑐) ⊆ (𝐹‘𝑑) ↔ (𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑑)))
24	21, 23	imbi12d 347	. . . . . . . 8 ⊢ (𝑐 = (𝑎 ∩ 𝑏) → ((𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ↔ ((𝑎 ∩ 𝑏) ⊆ 𝑑 → (𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑑))))
25		sseq2 3957	. . . . . . . . 9 ⊢ (𝑑 = 𝑎 → ((𝑎 ∩ 𝑏) ⊆ 𝑑 ↔ (𝑎 ∩ 𝑏) ⊆ 𝑎))
26		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑑 = 𝑎 → (𝐹‘𝑑) = (𝐹‘𝑎))
27	26	sseq2d 3963	. . . . . . . . 9 ⊢ (𝑑 = 𝑎 → ((𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑑) ↔ (𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑎)))
28	25, 27	imbi12d 347	. . . . . . . 8 ⊢ (𝑑 = 𝑎 → (((𝑎 ∩ 𝑏) ⊆ 𝑑 → (𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑑)) ↔ ((𝑎 ∩ 𝑏) ⊆ 𝑎 → (𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑎))))
29	24, 28	rspc2va 3588	. . . . . . 7 ⊢ ((((𝑎 ∩ 𝑏) ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ ∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑))) → ((𝑎 ∩ 𝑏) ⊆ 𝑎 → (𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑎)))
30	18, 19, 20, 29	syl21anc 851	. . . . . 6 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → ((𝑎 ∩ 𝑏) ⊆ 𝑎 → (𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑎)))
31	10, 30	mpi 21	. . . . 5 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑎))
32		simprr 785	. . . . . . 7 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → 𝑏 ∈ 𝒫 𝐴)
33		sseq2 3957	. . . . . . . . 9 ⊢ (𝑑 = 𝑏 → ((𝑎 ∩ 𝑏) ⊆ 𝑑 ↔ (𝑎 ∩ 𝑏) ⊆ 𝑏))
34		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑑 = 𝑏 → (𝐹‘𝑑) = (𝐹‘𝑏))
35	34	sseq2d 3963	. . . . . . . . 9 ⊢ (𝑑 = 𝑏 → ((𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑑) ↔ (𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑏)))
36	33, 35	imbi12d 347	. . . . . . . 8 ⊢ (𝑑 = 𝑏 → (((𝑎 ∩ 𝑏) ⊆ 𝑑 → (𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑑)) ↔ ((𝑎 ∩ 𝑏) ⊆ 𝑏 → (𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑏))))
37	24, 36	rspc2va 3588	. . . . . . 7 ⊢ ((((𝑎 ∩ 𝑏) ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) ∧ ∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑))) → ((𝑎 ∩ 𝑏) ⊆ 𝑏 → (𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑏)))
38	18, 32, 20, 37	syl21anc 851	. . . . . 6 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → ((𝑎 ∩ 𝑏) ⊆ 𝑏 → (𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑏)))
39	11, 38	mpi 21	. . . . 5 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝐹‘(𝑎 ∩ 𝑏)) ⊆ (𝐹‘𝑏))
40	31, 39	ssind 4186	. . . 4 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)))
41	40	ralrimivva 3206	. . 3 ⊢ (∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) → ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)))
42		dfss 3918	. . . . 5 ⊢ (𝑐 ⊆ 𝑑 ↔ 𝑐 = (𝑐 ∩ 𝑑))
43		fveq2 6885	. . . . . . . 8 ⊢ (𝑐 = (𝑐 ∩ 𝑑) → (𝐹‘𝑐) = (𝐹‘(𝑐 ∩ 𝑑)))
44	43	adantl 487	. . . . . . 7 ⊢ (((∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐 ∩ 𝑑)) → (𝐹‘𝑐) = (𝐹‘(𝑐 ∩ 𝑑)))
45		ineq1 4159	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑐 → (𝑎 ∩ 𝑏) = (𝑐 ∩ 𝑏))
46	45	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → (𝐹‘(𝑎 ∩ 𝑏)) = (𝐹‘(𝑐 ∩ 𝑏)))
47	2	ineq1d 4165	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → ((𝐹‘𝑎) ∩ (𝐹‘𝑏)) = ((𝐹‘𝑐) ∩ (𝐹‘𝑏)))
48	46, 47	sseq12d 3964	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑐 → ((𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)) ↔ (𝐹‘(𝑐 ∩ 𝑏)) ⊆ ((𝐹‘𝑐) ∩ (𝐹‘𝑏))))
49		ineq2 4160	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑑 → (𝑐 ∩ 𝑏) = (𝑐 ∩ 𝑑))
50	49	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑑 → (𝐹‘(𝑐 ∩ 𝑏)) = (𝐹‘(𝑐 ∩ 𝑑)))
51	6	ineq2d 4166	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑑 → ((𝐹‘𝑐) ∩ (𝐹‘𝑏)) = ((𝐹‘𝑐) ∩ (𝐹‘𝑑)))
52	50, 51	sseq12d 3964	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑑 → ((𝐹‘(𝑐 ∩ 𝑏)) ⊆ ((𝐹‘𝑐) ∩ (𝐹‘𝑏)) ↔ (𝐹‘(𝑐 ∩ 𝑑)) ⊆ ((𝐹‘𝑐) ∩ (𝐹‘𝑑))))
53	48, 52	rspc2va 3588	. . . . . . . . . 10 ⊢ (((𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴) ∧ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏))) → (𝐹‘(𝑐 ∩ 𝑑)) ⊆ ((𝐹‘𝑐) ∩ (𝐹‘𝑑)))
54	53	ancoms 464	. . . . . . . . 9 ⊢ ((∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴)) → (𝐹‘(𝑐 ∩ 𝑑)) ⊆ ((𝐹‘𝑐) ∩ (𝐹‘𝑑)))
55		inss2 4183	. . . . . . . . 9 ⊢ ((𝐹‘𝑐) ∩ (𝐹‘𝑑)) ⊆ (𝐹‘𝑑)
56	54, 55	sstrdi 3943	. . . . . . . 8 ⊢ ((∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴)) → (𝐹‘(𝑐 ∩ 𝑑)) ⊆ (𝐹‘𝑑))
57	56	adantr 486	. . . . . . 7 ⊢ (((∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐 ∩ 𝑑)) → (𝐹‘(𝑐 ∩ 𝑑)) ⊆ (𝐹‘𝑑))
58	44, 57	eqsstrd 3965	. . . . . 6 ⊢ (((∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐 ∩ 𝑑)) → (𝐹‘𝑐) ⊆ (𝐹‘𝑑))
59	58	ex 418	. . . . 5 ⊢ ((∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴)) → (𝑐 = (𝑐 ∩ 𝑑) → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)))
60	42, 59	biimtrid 245	. . . 4 ⊢ ((∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴)) → (𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)))
61	60	ralrimivva 3206	. . 3 ⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)) → ∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)))
62	41, 61	impbii 212	. 2 ⊢ (∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ↔ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)))
63	9, 62	bitri 278	1 ⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ‘cfv 6538

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-ext 2733  ax-sep 5249

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6494  df-fv 6546

This theorem is used by:  ntrk1k3eqk13  45049