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Theorem isotone2 41170
 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 𝐴𝑏 ∈ 𝒫 𝐴(𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
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.
StepHypRef Expression
1 sseq1 3919 . . . 4 (𝑎 = 𝑐 → (𝑎𝑏𝑐𝑏))
2 fveq2 6663 . . . . 5 (𝑎 = 𝑐 → (𝐹𝑎) = (𝐹𝑐))
32sseq1d 3925 . . . 4 (𝑎 = 𝑐 → ((𝐹𝑎) ⊆ (𝐹𝑏) ↔ (𝐹𝑐) ⊆ (𝐹𝑏)))
41, 3imbi12d 348 . . 3 (𝑎 = 𝑐 → ((𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) ↔ (𝑐𝑏 → (𝐹𝑐) ⊆ (𝐹𝑏))))
5 sseq2 3920 . . . 4 (𝑏 = 𝑑 → (𝑐𝑏𝑐𝑑))
6 fveq2 6663 . . . . 5 (𝑏 = 𝑑 → (𝐹𝑏) = (𝐹𝑑))
76sseq2d 3926 . . . 4 (𝑏 = 𝑑 → ((𝐹𝑐) ⊆ (𝐹𝑏) ↔ (𝐹𝑐) ⊆ (𝐹𝑑)))
85, 7imbi12d 348 . . 3 (𝑏 = 𝑑 → ((𝑐𝑏 → (𝐹𝑐) ⊆ (𝐹𝑏)) ↔ (𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑))))
94, 8cbvral2vw 3373 . 2 (∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) ↔ ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
10 inss1 4135 . . . . . 6 (𝑎𝑏) ⊆ 𝑎
11 inss2 4136 . . . . . . . . . 10 (𝑎𝑏) ⊆ 𝑏
12 elpwi 4506 . . . . . . . . . 10 (𝑏 ∈ 𝒫 𝐴𝑏𝐴)
1311, 12sstrid 3905 . . . . . . . . 9 (𝑏 ∈ 𝒫 𝐴 → (𝑎𝑏) ⊆ 𝐴)
14 vex 3413 . . . . . . . . . . 11 𝑏 ∈ V
1514inex2 5192 . . . . . . . . . 10 (𝑎𝑏) ∈ V
1615elpw 4501 . . . . . . . . 9 ((𝑎𝑏) ∈ 𝒫 𝐴 ↔ (𝑎𝑏) ⊆ 𝐴)
1713, 16sylibr 237 . . . . . . . 8 (𝑏 ∈ 𝒫 𝐴 → (𝑎𝑏) ∈ 𝒫 𝐴)
1817ad2antll 728 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎𝑏) ∈ 𝒫 𝐴)
19 simprl 770 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑎 ∈ 𝒫 𝐴)
20 simpl 486 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
21 sseq1 3919 . . . . . . . . 9 (𝑐 = (𝑎𝑏) → (𝑐𝑑 ↔ (𝑎𝑏) ⊆ 𝑑))
22 fveq2 6663 . . . . . . . . . 10 (𝑐 = (𝑎𝑏) → (𝐹𝑐) = (𝐹‘(𝑎𝑏)))
2322sseq1d 3925 . . . . . . . . 9 (𝑐 = (𝑎𝑏) → ((𝐹𝑐) ⊆ (𝐹𝑑) ↔ (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑)))
2421, 23imbi12d 348 . . . . . . . 8 (𝑐 = (𝑎𝑏) → ((𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ↔ ((𝑎𝑏) ⊆ 𝑑 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑))))
25 sseq2 3920 . . . . . . . . 9 (𝑑 = 𝑎 → ((𝑎𝑏) ⊆ 𝑑 ↔ (𝑎𝑏) ⊆ 𝑎))
26 fveq2 6663 . . . . . . . . . 10 (𝑑 = 𝑎 → (𝐹𝑑) = (𝐹𝑎))
2726sseq2d 3926 . . . . . . . . 9 (𝑑 = 𝑎 → ((𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑) ↔ (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎)))
2825, 27imbi12d 348 . . . . . . . 8 (𝑑 = 𝑎 → (((𝑎𝑏) ⊆ 𝑑 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑)) ↔ ((𝑎𝑏) ⊆ 𝑎 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎))))
2924, 28rspc2va 3554 . . . . . . 7 ((((𝑎𝑏) ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐴) ∧ ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑))) → ((𝑎𝑏) ⊆ 𝑎 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎)))
3018, 19, 20, 29syl21anc 836 . . . . . 6 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ((𝑎𝑏) ⊆ 𝑎 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎)))
3110, 30mpi 20 . . . . 5 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎))
32 simprr 772 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑏 ∈ 𝒫 𝐴)
33 sseq2 3920 . . . . . . . . 9 (𝑑 = 𝑏 → ((𝑎𝑏) ⊆ 𝑑 ↔ (𝑎𝑏) ⊆ 𝑏))
34 fveq2 6663 . . . . . . . . . 10 (𝑑 = 𝑏 → (𝐹𝑑) = (𝐹𝑏))
3534sseq2d 3926 . . . . . . . . 9 (𝑑 = 𝑏 → ((𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑) ↔ (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏)))
3633, 35imbi12d 348 . . . . . . . 8 (𝑑 = 𝑏 → (((𝑎𝑏) ⊆ 𝑑 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑)) ↔ ((𝑎𝑏) ⊆ 𝑏 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏))))
3724, 36rspc2va 3554 . . . . . . 7 ((((𝑎𝑏) ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑))) → ((𝑎𝑏) ⊆ 𝑏 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏)))
3818, 32, 20, 37syl21anc 836 . . . . . 6 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ((𝑎𝑏) ⊆ 𝑏 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏)))
3911, 38mpi 20 . . . . 5 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏))
4031, 39ssind 4139 . . . 4 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
4140ralrimivva 3120 . . 3 (∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) → ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
42 dfss 3878 . . . . 5 (𝑐𝑑𝑐 = (𝑐𝑑))
43 fveq2 6663 . . . . . . . 8 (𝑐 = (𝑐𝑑) → (𝐹𝑐) = (𝐹‘(𝑐𝑑)))
4443adantl 485 . . . . . . 7 (((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐𝑑)) → (𝐹𝑐) = (𝐹‘(𝑐𝑑)))
45 ineq1 4111 . . . . . . . . . . . . 13 (𝑎 = 𝑐 → (𝑎𝑏) = (𝑐𝑏))
4645fveq2d 6667 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝐹‘(𝑎𝑏)) = (𝐹‘(𝑐𝑏)))
472ineq1d 4118 . . . . . . . . . . . 12 (𝑎 = 𝑐 → ((𝐹𝑎) ∩ (𝐹𝑏)) = ((𝐹𝑐) ∩ (𝐹𝑏)))
4846, 47sseq12d 3927 . . . . . . . . . . 11 (𝑎 = 𝑐 → ((𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ↔ (𝐹‘(𝑐𝑏)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑏))))
49 ineq2 4113 . . . . . . . . . . . . 13 (𝑏 = 𝑑 → (𝑐𝑏) = (𝑐𝑑))
5049fveq2d 6667 . . . . . . . . . . . 12 (𝑏 = 𝑑 → (𝐹‘(𝑐𝑏)) = (𝐹‘(𝑐𝑑)))
516ineq2d 4119 . . . . . . . . . . . 12 (𝑏 = 𝑑 → ((𝐹𝑐) ∩ (𝐹𝑏)) = ((𝐹𝑐) ∩ (𝐹𝑑)))
5250, 51sseq12d 3927 . . . . . . . . . . 11 (𝑏 = 𝑑 → ((𝐹‘(𝑐𝑏)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑏)) ↔ (𝐹‘(𝑐𝑑)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑑))))
5348, 52rspc2va 3554 . . . . . . . . . 10 (((𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴) ∧ ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏))) → (𝐹‘(𝑐𝑑)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑑)))
5453ancoms 462 . . . . . . . . 9 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝐹‘(𝑐𝑑)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑑)))
55 inss2 4136 . . . . . . . . 9 ((𝐹𝑐) ∩ (𝐹𝑑)) ⊆ (𝐹𝑑)
5654, 55sstrdi 3906 . . . . . . . 8 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝐹‘(𝑐𝑑)) ⊆ (𝐹𝑑))
5756adantr 484 . . . . . . 7 (((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐𝑑)) → (𝐹‘(𝑐𝑑)) ⊆ (𝐹𝑑))
5844, 57eqsstrd 3932 . . . . . 6 (((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐𝑑)) → (𝐹𝑐) ⊆ (𝐹𝑑))
5958ex 416 . . . . 5 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝑐 = (𝑐𝑑) → (𝐹𝑐) ⊆ (𝐹𝑑)))
6042, 59syl5bi 245 . . . 4 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
6160ralrimivva 3120 . . 3 (∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) → ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
6241, 61impbii 212 . 2 (∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ↔ ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
639, 62bitri 278 1 (∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070   ∩ cin 3859   ⊆ wss 3860  𝒫 cpw 4497  ‘cfv 6340 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5173 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rab 3079  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-iota 6299  df-fv 6348 This theorem is referenced by:  ntrk1k3eqk13  41171
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