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Theorem isotone2 40261
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 3995 . . . 4 (𝑎 = 𝑐 → (𝑎𝑏𝑐𝑏))
2 fveq2 6666 . . . . 5 (𝑎 = 𝑐 → (𝐹𝑎) = (𝐹𝑐))
32sseq1d 4001 . . . 4 (𝑎 = 𝑐 → ((𝐹𝑎) ⊆ (𝐹𝑏) ↔ (𝐹𝑐) ⊆ (𝐹𝑏)))
41, 3imbi12d 346 . . 3 (𝑎 = 𝑐 → ((𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) ↔ (𝑐𝑏 → (𝐹𝑐) ⊆ (𝐹𝑏))))
5 sseq2 3996 . . . 4 (𝑏 = 𝑑 → (𝑐𝑏𝑐𝑑))
6 fveq2 6666 . . . . 5 (𝑏 = 𝑑 → (𝐹𝑏) = (𝐹𝑑))
76sseq2d 4002 . . . 4 (𝑏 = 𝑑 → ((𝐹𝑐) ⊆ (𝐹𝑏) ↔ (𝐹𝑐) ⊆ (𝐹𝑑)))
85, 7imbi12d 346 . . 3 (𝑏 = 𝑑 → ((𝑐𝑏 → (𝐹𝑐) ⊆ (𝐹𝑏)) ↔ (𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑))))
94, 8cbvral2v 3469 . 2 (∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) ↔ ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
10 inss1 4208 . . . . . 6 (𝑎𝑏) ⊆ 𝑎
11 inss2 4209 . . . . . . . . . 10 (𝑎𝑏) ⊆ 𝑏
12 elpwi 4553 . . . . . . . . . 10 (𝑏 ∈ 𝒫 𝐴𝑏𝐴)
1311, 12sstrid 3981 . . . . . . . . 9 (𝑏 ∈ 𝒫 𝐴 → (𝑎𝑏) ⊆ 𝐴)
14 vex 3502 . . . . . . . . . . 11 𝑏 ∈ V
1514inex2 5218 . . . . . . . . . 10 (𝑎𝑏) ∈ V
1615elpw 4548 . . . . . . . . 9 ((𝑎𝑏) ∈ 𝒫 𝐴 ↔ (𝑎𝑏) ⊆ 𝐴)
1713, 16sylibr 235 . . . . . . . 8 (𝑏 ∈ 𝒫 𝐴 → (𝑎𝑏) ∈ 𝒫 𝐴)
1817ad2antll 725 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎𝑏) ∈ 𝒫 𝐴)
19 simprl 767 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑎 ∈ 𝒫 𝐴)
20 simpl 483 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
21 sseq1 3995 . . . . . . . . 9 (𝑐 = (𝑎𝑏) → (𝑐𝑑 ↔ (𝑎𝑏) ⊆ 𝑑))
22 fveq2 6666 . . . . . . . . . 10 (𝑐 = (𝑎𝑏) → (𝐹𝑐) = (𝐹‘(𝑎𝑏)))
2322sseq1d 4001 . . . . . . . . 9 (𝑐 = (𝑎𝑏) → ((𝐹𝑐) ⊆ (𝐹𝑑) ↔ (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑)))
2421, 23imbi12d 346 . . . . . . . 8 (𝑐 = (𝑎𝑏) → ((𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ↔ ((𝑎𝑏) ⊆ 𝑑 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑))))
25 sseq2 3996 . . . . . . . . 9 (𝑑 = 𝑎 → ((𝑎𝑏) ⊆ 𝑑 ↔ (𝑎𝑏) ⊆ 𝑎))
26 fveq2 6666 . . . . . . . . . 10 (𝑑 = 𝑎 → (𝐹𝑑) = (𝐹𝑎))
2726sseq2d 4002 . . . . . . . . 9 (𝑑 = 𝑎 → ((𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑) ↔ (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎)))
2825, 27imbi12d 346 . . . . . . . 8 (𝑑 = 𝑎 → (((𝑎𝑏) ⊆ 𝑑 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑)) ↔ ((𝑎𝑏) ⊆ 𝑎 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎))))
2924, 28rspc2va 3637 . . . . . . 7 ((((𝑎𝑏) ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐴) ∧ ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑))) → ((𝑎𝑏) ⊆ 𝑎 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎)))
3018, 19, 20, 29syl21anc 835 . . . . . 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 769 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑏 ∈ 𝒫 𝐴)
33 sseq2 3996 . . . . . . . . 9 (𝑑 = 𝑏 → ((𝑎𝑏) ⊆ 𝑑 ↔ (𝑎𝑏) ⊆ 𝑏))
34 fveq2 6666 . . . . . . . . . 10 (𝑑 = 𝑏 → (𝐹𝑑) = (𝐹𝑏))
3534sseq2d 4002 . . . . . . . . 9 (𝑑 = 𝑏 → ((𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑) ↔ (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏)))
3633, 35imbi12d 346 . . . . . . . 8 (𝑑 = 𝑏 → (((𝑎𝑏) ⊆ 𝑑 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑)) ↔ ((𝑎𝑏) ⊆ 𝑏 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏))))
3724, 36rspc2va 3637 . . . . . . 7 ((((𝑎𝑏) ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑))) → ((𝑎𝑏) ⊆ 𝑏 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏)))
3818, 32, 20, 37syl21anc 835 . . . . . 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 4212 . . . 4 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
4140ralrimivva 3195 . . 3 (∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) → ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
42 dfss 3956 . . . . 5 (𝑐𝑑𝑐 = (𝑐𝑑))
43 fveq2 6666 . . . . . . . 8 (𝑐 = (𝑐𝑑) → (𝐹𝑐) = (𝐹‘(𝑐𝑑)))
4443adantl 482 . . . . . . 7 (((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐𝑑)) → (𝐹𝑐) = (𝐹‘(𝑐𝑑)))
45 ineq1 4184 . . . . . . . . . . . . 13 (𝑎 = 𝑐 → (𝑎𝑏) = (𝑐𝑏))
4645fveq2d 6670 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝐹‘(𝑎𝑏)) = (𝐹‘(𝑐𝑏)))
472ineq1d 4191 . . . . . . . . . . . 12 (𝑎 = 𝑐 → ((𝐹𝑎) ∩ (𝐹𝑏)) = ((𝐹𝑐) ∩ (𝐹𝑏)))
4846, 47sseq12d 4003 . . . . . . . . . . 11 (𝑎 = 𝑐 → ((𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ↔ (𝐹‘(𝑐𝑏)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑏))))
49 ineq2 4186 . . . . . . . . . . . . 13 (𝑏 = 𝑑 → (𝑐𝑏) = (𝑐𝑑))
5049fveq2d 6670 . . . . . . . . . . . 12 (𝑏 = 𝑑 → (𝐹‘(𝑐𝑏)) = (𝐹‘(𝑐𝑑)))
516ineq2d 4192 . . . . . . . . . . . 12 (𝑏 = 𝑑 → ((𝐹𝑐) ∩ (𝐹𝑏)) = ((𝐹𝑐) ∩ (𝐹𝑑)))
5250, 51sseq12d 4003 . . . . . . . . . . 11 (𝑏 = 𝑑 → ((𝐹‘(𝑐𝑏)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑏)) ↔ (𝐹‘(𝑐𝑑)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑑))))
5348, 52rspc2va 3637 . . . . . . . . . 10 (((𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴) ∧ ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏))) → (𝐹‘(𝑐𝑑)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑑)))
5453ancoms 459 . . . . . . . . 9 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝐹‘(𝑐𝑑)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑑)))
55 inss2 4209 . . . . . . . . 9 ((𝐹𝑐) ∩ (𝐹𝑑)) ⊆ (𝐹𝑑)
5654, 55syl6ss 3982 . . . . . . . 8 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝐹‘(𝑐𝑑)) ⊆ (𝐹𝑑))
5756adantr 481 . . . . . . 7 (((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐𝑑)) → (𝐹‘(𝑐𝑑)) ⊆ (𝐹𝑑))
5844, 57eqsstrd 4008 . . . . . 6 (((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐𝑑)) → (𝐹𝑐) ⊆ (𝐹𝑑))
5958ex 413 . . . . 5 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝑐 = (𝑐𝑑) → (𝐹𝑐) ⊆ (𝐹𝑑)))
6042, 59syl5bi 243 . . . 4 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
6160ralrimivva 3195 . . 3 (∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) → ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
6241, 61impbii 210 . 2 (∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ↔ ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
639, 62bitri 276 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 207  wa 396   = wceq 1530  wcel 2106  wral 3142  cin 3938  wss 3939  𝒫 cpw 4541  cfv 6351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-13 2385  ax-ext 2796  ax-sep 5199
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-iota 6311  df-fv 6359
This theorem is referenced by:  ntrk1k3eqk13  40262
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