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Theorem isotone2 42395
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 3974 . . . 4 (𝑎 = 𝑐 → (𝑎𝑏𝑐𝑏))
2 fveq2 6847 . . . . 5 (𝑎 = 𝑐 → (𝐹𝑎) = (𝐹𝑐))
32sseq1d 3980 . . . 4 (𝑎 = 𝑐 → ((𝐹𝑎) ⊆ (𝐹𝑏) ↔ (𝐹𝑐) ⊆ (𝐹𝑏)))
41, 3imbi12d 345 . . 3 (𝑎 = 𝑐 → ((𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) ↔ (𝑐𝑏 → (𝐹𝑐) ⊆ (𝐹𝑏))))
5 sseq2 3975 . . . 4 (𝑏 = 𝑑 → (𝑐𝑏𝑐𝑑))
6 fveq2 6847 . . . . 5 (𝑏 = 𝑑 → (𝐹𝑏) = (𝐹𝑑))
76sseq2d 3981 . . . 4 (𝑏 = 𝑑 → ((𝐹𝑐) ⊆ (𝐹𝑏) ↔ (𝐹𝑐) ⊆ (𝐹𝑑)))
85, 7imbi12d 345 . . 3 (𝑏 = 𝑑 → ((𝑐𝑏 → (𝐹𝑐) ⊆ (𝐹𝑏)) ↔ (𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑))))
94, 8cbvral2vw 3230 . 2 (∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) ↔ ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
10 inss1 4193 . . . . . 6 (𝑎𝑏) ⊆ 𝑎
11 inss2 4194 . . . . . . . . . 10 (𝑎𝑏) ⊆ 𝑏
12 elpwi 4572 . . . . . . . . . 10 (𝑏 ∈ 𝒫 𝐴𝑏𝐴)
1311, 12sstrid 3960 . . . . . . . . 9 (𝑏 ∈ 𝒫 𝐴 → (𝑎𝑏) ⊆ 𝐴)
14 vex 3452 . . . . . . . . . . 11 𝑏 ∈ V
1514inex2 5280 . . . . . . . . . 10 (𝑎𝑏) ∈ V
1615elpw 4569 . . . . . . . . 9 ((𝑎𝑏) ∈ 𝒫 𝐴 ↔ (𝑎𝑏) ⊆ 𝐴)
1713, 16sylibr 233 . . . . . . . 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 484 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
21 sseq1 3974 . . . . . . . . 9 (𝑐 = (𝑎𝑏) → (𝑐𝑑 ↔ (𝑎𝑏) ⊆ 𝑑))
22 fveq2 6847 . . . . . . . . . 10 (𝑐 = (𝑎𝑏) → (𝐹𝑐) = (𝐹‘(𝑎𝑏)))
2322sseq1d 3980 . . . . . . . . 9 (𝑐 = (𝑎𝑏) → ((𝐹𝑐) ⊆ (𝐹𝑑) ↔ (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑)))
2421, 23imbi12d 345 . . . . . . . 8 (𝑐 = (𝑎𝑏) → ((𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ↔ ((𝑎𝑏) ⊆ 𝑑 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑))))
25 sseq2 3975 . . . . . . . . 9 (𝑑 = 𝑎 → ((𝑎𝑏) ⊆ 𝑑 ↔ (𝑎𝑏) ⊆ 𝑎))
26 fveq2 6847 . . . . . . . . . 10 (𝑑 = 𝑎 → (𝐹𝑑) = (𝐹𝑎))
2726sseq2d 3981 . . . . . . . . 9 (𝑑 = 𝑎 → ((𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑) ↔ (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎)))
2825, 27imbi12d 345 . . . . . . . 8 (𝑑 = 𝑎 → (((𝑎𝑏) ⊆ 𝑑 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑)) ↔ ((𝑎𝑏) ⊆ 𝑎 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎))))
2924, 28rspc2va 3594 . . . . . . 7 ((((𝑎𝑏) ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐴) ∧ ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑))) → ((𝑎𝑏) ⊆ 𝑎 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎)))
3018, 19, 20, 29syl21anc 837 . . . . . 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 3975 . . . . . . . . 9 (𝑑 = 𝑏 → ((𝑎𝑏) ⊆ 𝑑 ↔ (𝑎𝑏) ⊆ 𝑏))
34 fveq2 6847 . . . . . . . . . 10 (𝑑 = 𝑏 → (𝐹𝑑) = (𝐹𝑏))
3534sseq2d 3981 . . . . . . . . 9 (𝑑 = 𝑏 → ((𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑) ↔ (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏)))
3633, 35imbi12d 345 . . . . . . . 8 (𝑑 = 𝑏 → (((𝑎𝑏) ⊆ 𝑑 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑)) ↔ ((𝑎𝑏) ⊆ 𝑏 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏))))
3724, 36rspc2va 3594 . . . . . . 7 ((((𝑎𝑏) ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑))) → ((𝑎𝑏) ⊆ 𝑏 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏)))
3818, 32, 20, 37syl21anc 837 . . . . . 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 4197 . . . 4 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
4140ralrimivva 3198 . . 3 (∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) → ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
42 dfss 3933 . . . . 5 (𝑐𝑑𝑐 = (𝑐𝑑))
43 fveq2 6847 . . . . . . . 8 (𝑐 = (𝑐𝑑) → (𝐹𝑐) = (𝐹‘(𝑐𝑑)))
4443adantl 483 . . . . . . 7 (((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐𝑑)) → (𝐹𝑐) = (𝐹‘(𝑐𝑑)))
45 ineq1 4170 . . . . . . . . . . . . 13 (𝑎 = 𝑐 → (𝑎𝑏) = (𝑐𝑏))
4645fveq2d 6851 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝐹‘(𝑎𝑏)) = (𝐹‘(𝑐𝑏)))
472ineq1d 4176 . . . . . . . . . . . 12 (𝑎 = 𝑐 → ((𝐹𝑎) ∩ (𝐹𝑏)) = ((𝐹𝑐) ∩ (𝐹𝑏)))
4846, 47sseq12d 3982 . . . . . . . . . . 11 (𝑎 = 𝑐 → ((𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ↔ (𝐹‘(𝑐𝑏)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑏))))
49 ineq2 4171 . . . . . . . . . . . . 13 (𝑏 = 𝑑 → (𝑐𝑏) = (𝑐𝑑))
5049fveq2d 6851 . . . . . . . . . . . 12 (𝑏 = 𝑑 → (𝐹‘(𝑐𝑏)) = (𝐹‘(𝑐𝑑)))
516ineq2d 4177 . . . . . . . . . . . 12 (𝑏 = 𝑑 → ((𝐹𝑐) ∩ (𝐹𝑏)) = ((𝐹𝑐) ∩ (𝐹𝑑)))
5250, 51sseq12d 3982 . . . . . . . . . . 11 (𝑏 = 𝑑 → ((𝐹‘(𝑐𝑏)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑏)) ↔ (𝐹‘(𝑐𝑑)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑑))))
5348, 52rspc2va 3594 . . . . . . . . . 10 (((𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴) ∧ ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏))) → (𝐹‘(𝑐𝑑)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑑)))
5453ancoms 460 . . . . . . . . 9 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝐹‘(𝑐𝑑)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑑)))
55 inss2 4194 . . . . . . . . 9 ((𝐹𝑐) ∩ (𝐹𝑑)) ⊆ (𝐹𝑑)
5654, 55sstrdi 3961 . . . . . . . 8 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝐹‘(𝑐𝑑)) ⊆ (𝐹𝑑))
5756adantr 482 . . . . . . 7 (((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐𝑑)) → (𝐹‘(𝑐𝑑)) ⊆ (𝐹𝑑))
5844, 57eqsstrd 3987 . . . . . 6 (((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐𝑑)) → (𝐹𝑐) ⊆ (𝐹𝑑))
5958ex 414 . . . . 5 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝑐 = (𝑐𝑑) → (𝐹𝑐) ⊆ (𝐹𝑑)))
6042, 59biimtrid 241 . . . 4 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
6160ralrimivva 3198 . . 3 (∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) → ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
6241, 61impbii 208 . 2 (∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ↔ ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
639, 62bitri 275 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 205  wa 397   = wceq 1542  wcel 2107  wral 3065  cin 3914  wss 3915  𝒫 cpw 4565  cfv 6501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509
This theorem is referenced by:  ntrk1k3eqk13  42396
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