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Theorem isotone2 38867
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 3767 . . . 4 (𝑎 = 𝑐 → (𝑎𝑏𝑐𝑏))
2 fveq2 6353 . . . . 5 (𝑎 = 𝑐 → (𝐹𝑎) = (𝐹𝑐))
32sseq1d 3773 . . . 4 (𝑎 = 𝑐 → ((𝐹𝑎) ⊆ (𝐹𝑏) ↔ (𝐹𝑐) ⊆ (𝐹𝑏)))
41, 3imbi12d 333 . . 3 (𝑎 = 𝑐 → ((𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) ↔ (𝑐𝑏 → (𝐹𝑐) ⊆ (𝐹𝑏))))
5 sseq2 3768 . . . 4 (𝑏 = 𝑑 → (𝑐𝑏𝑐𝑑))
6 fveq2 6353 . . . . 5 (𝑏 = 𝑑 → (𝐹𝑏) = (𝐹𝑑))
76sseq2d 3774 . . . 4 (𝑏 = 𝑑 → ((𝐹𝑐) ⊆ (𝐹𝑏) ↔ (𝐹𝑐) ⊆ (𝐹𝑑)))
85, 7imbi12d 333 . . 3 (𝑏 = 𝑑 → ((𝑐𝑏 → (𝐹𝑐) ⊆ (𝐹𝑏)) ↔ (𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑))))
94, 8cbvral2v 3318 . 2 (∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) ↔ ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
10 inss1 3976 . . . . . 6 (𝑎𝑏) ⊆ 𝑎
11 inss2 3977 . . . . . . . . . 10 (𝑎𝑏) ⊆ 𝑏
12 elpwi 4312 . . . . . . . . . 10 (𝑏 ∈ 𝒫 𝐴𝑏𝐴)
1311, 12syl5ss 3755 . . . . . . . . 9 (𝑏 ∈ 𝒫 𝐴 → (𝑎𝑏) ⊆ 𝐴)
14 vex 3343 . . . . . . . . . . 11 𝑏 ∈ V
1514inex2 4952 . . . . . . . . . 10 (𝑎𝑏) ∈ V
1615elpw 4308 . . . . . . . . 9 ((𝑎𝑏) ∈ 𝒫 𝐴 ↔ (𝑎𝑏) ⊆ 𝐴)
1713, 16sylibr 224 . . . . . . . 8 (𝑏 ∈ 𝒫 𝐴 → (𝑎𝑏) ∈ 𝒫 𝐴)
1817ad2antll 767 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎𝑏) ∈ 𝒫 𝐴)
19 simprl 811 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑎 ∈ 𝒫 𝐴)
20 simpl 474 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
21 sseq1 3767 . . . . . . . . 9 (𝑐 = (𝑎𝑏) → (𝑐𝑑 ↔ (𝑎𝑏) ⊆ 𝑑))
22 fveq2 6353 . . . . . . . . . 10 (𝑐 = (𝑎𝑏) → (𝐹𝑐) = (𝐹‘(𝑎𝑏)))
2322sseq1d 3773 . . . . . . . . 9 (𝑐 = (𝑎𝑏) → ((𝐹𝑐) ⊆ (𝐹𝑑) ↔ (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑)))
2421, 23imbi12d 333 . . . . . . . 8 (𝑐 = (𝑎𝑏) → ((𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ↔ ((𝑎𝑏) ⊆ 𝑑 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑))))
25 sseq2 3768 . . . . . . . . 9 (𝑑 = 𝑎 → ((𝑎𝑏) ⊆ 𝑑 ↔ (𝑎𝑏) ⊆ 𝑎))
26 fveq2 6353 . . . . . . . . . 10 (𝑑 = 𝑎 → (𝐹𝑑) = (𝐹𝑎))
2726sseq2d 3774 . . . . . . . . 9 (𝑑 = 𝑎 → ((𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑) ↔ (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎)))
2825, 27imbi12d 333 . . . . . . . 8 (𝑑 = 𝑎 → (((𝑎𝑏) ⊆ 𝑑 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑)) ↔ ((𝑎𝑏) ⊆ 𝑎 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎))))
2924, 28rspc2va 3462 . . . . . . 7 ((((𝑎𝑏) ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐴) ∧ ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑))) → ((𝑎𝑏) ⊆ 𝑎 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎)))
3018, 19, 20, 29syl21anc 1476 . . . . . 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 813 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑏 ∈ 𝒫 𝐴)
33 sseq2 3768 . . . . . . . . 9 (𝑑 = 𝑏 → ((𝑎𝑏) ⊆ 𝑑 ↔ (𝑎𝑏) ⊆ 𝑏))
34 fveq2 6353 . . . . . . . . . 10 (𝑑 = 𝑏 → (𝐹𝑑) = (𝐹𝑏))
3534sseq2d 3774 . . . . . . . . 9 (𝑑 = 𝑏 → ((𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑) ↔ (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏)))
3633, 35imbi12d 333 . . . . . . . 8 (𝑑 = 𝑏 → (((𝑎𝑏) ⊆ 𝑑 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑)) ↔ ((𝑎𝑏) ⊆ 𝑏 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏))))
3724, 36rspc2va 3462 . . . . . . 7 ((((𝑎𝑏) ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑))) → ((𝑎𝑏) ⊆ 𝑏 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏)))
3818, 32, 20, 37syl21anc 1476 . . . . . 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 3980 . . . 4 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
4140ralrimivva 3109 . . 3 (∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) → ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
42 dfss 3730 . . . . 5 (𝑐𝑑𝑐 = (𝑐𝑑))
43 fveq2 6353 . . . . . . . 8 (𝑐 = (𝑐𝑑) → (𝐹𝑐) = (𝐹‘(𝑐𝑑)))
4443adantl 473 . . . . . . 7 (((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐𝑑)) → (𝐹𝑐) = (𝐹‘(𝑐𝑑)))
45 ineq1 3950 . . . . . . . . . . . . 13 (𝑎 = 𝑐 → (𝑎𝑏) = (𝑐𝑏))
4645fveq2d 6357 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝐹‘(𝑎𝑏)) = (𝐹‘(𝑐𝑏)))
472ineq1d 3956 . . . . . . . . . . . 12 (𝑎 = 𝑐 → ((𝐹𝑎) ∩ (𝐹𝑏)) = ((𝐹𝑐) ∩ (𝐹𝑏)))
4846, 47sseq12d 3775 . . . . . . . . . . 11 (𝑎 = 𝑐 → ((𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ↔ (𝐹‘(𝑐𝑏)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑏))))
49 ineq2 3951 . . . . . . . . . . . . 13 (𝑏 = 𝑑 → (𝑐𝑏) = (𝑐𝑑))
5049fveq2d 6357 . . . . . . . . . . . 12 (𝑏 = 𝑑 → (𝐹‘(𝑐𝑏)) = (𝐹‘(𝑐𝑑)))
516ineq2d 3957 . . . . . . . . . . . 12 (𝑏 = 𝑑 → ((𝐹𝑐) ∩ (𝐹𝑏)) = ((𝐹𝑐) ∩ (𝐹𝑑)))
5250, 51sseq12d 3775 . . . . . . . . . . 11 (𝑏 = 𝑑 → ((𝐹‘(𝑐𝑏)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑏)) ↔ (𝐹‘(𝑐𝑑)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑑))))
5348, 52rspc2va 3462 . . . . . . . . . 10 (((𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴) ∧ ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏))) → (𝐹‘(𝑐𝑑)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑑)))
5453ancoms 468 . . . . . . . . 9 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝐹‘(𝑐𝑑)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑑)))
55 inss2 3977 . . . . . . . . 9 ((𝐹𝑐) ∩ (𝐹𝑑)) ⊆ (𝐹𝑑)
5654, 55syl6ss 3756 . . . . . . . 8 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝐹‘(𝑐𝑑)) ⊆ (𝐹𝑑))
5756adantr 472 . . . . . . 7 (((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐𝑑)) → (𝐹‘(𝑐𝑑)) ⊆ (𝐹𝑑))
5844, 57eqsstrd 3780 . . . . . 6 (((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐𝑑)) → (𝐹𝑐) ⊆ (𝐹𝑑))
5958ex 449 . . . . 5 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝑐 = (𝑐𝑑) → (𝐹𝑐) ⊆ (𝐹𝑑)))
6042, 59syl5bi 232 . . . 4 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
6160ralrimivva 3109 . . 3 (∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) → ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
6241, 61impbii 199 . 2 (∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ↔ ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
639, 62bitri 264 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 196  wa 383   = wceq 1632  wcel 2139  wral 3050  cin 3714  wss 3715  𝒫 cpw 4302  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057
This theorem is referenced by:  ntrk1k3eqk13  38868
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