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Theorem isotone2 37868
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 3611 . . . 4 (𝑎 = 𝑐 → (𝑎𝑏𝑐𝑏))
2 fveq2 6158 . . . . 5 (𝑎 = 𝑐 → (𝐹𝑎) = (𝐹𝑐))
32sseq1d 3617 . . . 4 (𝑎 = 𝑐 → ((𝐹𝑎) ⊆ (𝐹𝑏) ↔ (𝐹𝑐) ⊆ (𝐹𝑏)))
41, 3imbi12d 334 . . 3 (𝑎 = 𝑐 → ((𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) ↔ (𝑐𝑏 → (𝐹𝑐) ⊆ (𝐹𝑏))))
5 sseq2 3612 . . . 4 (𝑏 = 𝑑 → (𝑐𝑏𝑐𝑑))
6 fveq2 6158 . . . . 5 (𝑏 = 𝑑 → (𝐹𝑏) = (𝐹𝑑))
76sseq2d 3618 . . . 4 (𝑏 = 𝑑 → ((𝐹𝑐) ⊆ (𝐹𝑏) ↔ (𝐹𝑐) ⊆ (𝐹𝑑)))
85, 7imbi12d 334 . . 3 (𝑏 = 𝑑 → ((𝑐𝑏 → (𝐹𝑐) ⊆ (𝐹𝑏)) ↔ (𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑))))
94, 8cbvral2v 3171 . 2 (∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) ↔ ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
10 inss1 3817 . . . . . 6 (𝑎𝑏) ⊆ 𝑎
11 inss2 3818 . . . . . . . . . 10 (𝑎𝑏) ⊆ 𝑏
12 elpwi 4146 . . . . . . . . . 10 (𝑏 ∈ 𝒫 𝐴𝑏𝐴)
1311, 12syl5ss 3599 . . . . . . . . 9 (𝑏 ∈ 𝒫 𝐴 → (𝑎𝑏) ⊆ 𝐴)
14 vex 3193 . . . . . . . . . . 11 𝑏 ∈ V
1514inex2 4770 . . . . . . . . . 10 (𝑎𝑏) ∈ V
1615elpw 4142 . . . . . . . . 9 ((𝑎𝑏) ∈ 𝒫 𝐴 ↔ (𝑎𝑏) ⊆ 𝐴)
1713, 16sylibr 224 . . . . . . . 8 (𝑏 ∈ 𝒫 𝐴 → (𝑎𝑏) ∈ 𝒫 𝐴)
1817ad2antll 764 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎𝑏) ∈ 𝒫 𝐴)
19 simprl 793 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑎 ∈ 𝒫 𝐴)
20 simpl 473 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)))
21 sseq1 3611 . . . . . . . . 9 (𝑐 = (𝑎𝑏) → (𝑐𝑑 ↔ (𝑎𝑏) ⊆ 𝑑))
22 fveq2 6158 . . . . . . . . . 10 (𝑐 = (𝑎𝑏) → (𝐹𝑐) = (𝐹‘(𝑎𝑏)))
2322sseq1d 3617 . . . . . . . . 9 (𝑐 = (𝑎𝑏) → ((𝐹𝑐) ⊆ (𝐹𝑑) ↔ (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑)))
2421, 23imbi12d 334 . . . . . . . 8 (𝑐 = (𝑎𝑏) → ((𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ↔ ((𝑎𝑏) ⊆ 𝑑 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑))))
25 sseq2 3612 . . . . . . . . 9 (𝑑 = 𝑎 → ((𝑎𝑏) ⊆ 𝑑 ↔ (𝑎𝑏) ⊆ 𝑎))
26 fveq2 6158 . . . . . . . . . 10 (𝑑 = 𝑎 → (𝐹𝑑) = (𝐹𝑎))
2726sseq2d 3618 . . . . . . . . 9 (𝑑 = 𝑎 → ((𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑) ↔ (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎)))
2825, 27imbi12d 334 . . . . . . . 8 (𝑑 = 𝑎 → (((𝑎𝑏) ⊆ 𝑑 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑)) ↔ ((𝑎𝑏) ⊆ 𝑎 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎))))
2924, 28rspc2va 3312 . . . . . . 7 ((((𝑎𝑏) ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐴) ∧ ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑))) → ((𝑎𝑏) ⊆ 𝑎 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑎)))
3018, 19, 20, 29syl21anc 1322 . . . . . 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 795 . . . . . . 7 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑏 ∈ 𝒫 𝐴)
33 sseq2 3612 . . . . . . . . 9 (𝑑 = 𝑏 → ((𝑎𝑏) ⊆ 𝑑 ↔ (𝑎𝑏) ⊆ 𝑏))
34 fveq2 6158 . . . . . . . . . 10 (𝑑 = 𝑏 → (𝐹𝑑) = (𝐹𝑏))
3534sseq2d 3618 . . . . . . . . 9 (𝑑 = 𝑏 → ((𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑) ↔ (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏)))
3633, 35imbi12d 334 . . . . . . . 8 (𝑑 = 𝑏 → (((𝑎𝑏) ⊆ 𝑑 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑑)) ↔ ((𝑎𝑏) ⊆ 𝑏 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏))))
3724, 36rspc2va 3312 . . . . . . 7 ((((𝑎𝑏) ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ ∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑))) → ((𝑎𝑏) ⊆ 𝑏 → (𝐹‘(𝑎𝑏)) ⊆ (𝐹𝑏)))
3818, 32, 20, 37syl21anc 1322 . . . . . 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 3821 . . . 4 ((∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
4140ralrimivva 2967 . . 3 (∀𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴(𝑐𝑑 → (𝐹𝑐) ⊆ (𝐹𝑑)) → ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))
42 dfss 3575 . . . . 5 (𝑐𝑑𝑐 = (𝑐𝑑))
43 fveq2 6158 . . . . . . . 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 3791 . . . . . . . . . . . . 13 (𝑎 = 𝑐 → (𝑎𝑏) = (𝑐𝑏))
4645fveq2d 6162 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝐹‘(𝑎𝑏)) = (𝐹‘(𝑐𝑏)))
472ineq1d 3797 . . . . . . . . . . . 12 (𝑎 = 𝑐 → ((𝐹𝑎) ∩ (𝐹𝑏)) = ((𝐹𝑐) ∩ (𝐹𝑏)))
4846, 47sseq12d 3619 . . . . . . . . . . 11 (𝑎 = 𝑐 → ((𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ↔ (𝐹‘(𝑐𝑏)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑏))))
49 ineq2 3792 . . . . . . . . . . . . 13 (𝑏 = 𝑑 → (𝑐𝑏) = (𝑐𝑑))
5049fveq2d 6162 . . . . . . . . . . . 12 (𝑏 = 𝑑 → (𝐹‘(𝑐𝑏)) = (𝐹‘(𝑐𝑑)))
516ineq2d 3798 . . . . . . . . . . . 12 (𝑏 = 𝑑 → ((𝐹𝑐) ∩ (𝐹𝑏)) = ((𝐹𝑐) ∩ (𝐹𝑑)))
5250, 51sseq12d 3619 . . . . . . . . . . 11 (𝑏 = 𝑑 → ((𝐹‘(𝑐𝑏)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑏)) ↔ (𝐹‘(𝑐𝑑)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑑))))
5348, 52rspc2va 3312 . . . . . . . . . 10 (((𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴) ∧ ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏))) → (𝐹‘(𝑐𝑑)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑑)))
5453ancoms 469 . . . . . . . . 9 ((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) → (𝐹‘(𝑐𝑑)) ⊆ ((𝐹𝑐) ∩ (𝐹𝑑)))
55 inss2 3818 . . . . . . . . 9 ((𝐹𝑐) ∩ (𝐹𝑑)) ⊆ (𝐹𝑑)
5654, 55syl6ss 3600 . . . . . . . 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 3624 . . . . . 6 (((∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴𝑑 ∈ 𝒫 𝐴)) ∧ 𝑐 = (𝑐𝑑)) → (𝐹𝑐) ⊆ (𝐹𝑑))
5958ex 450 . . . . 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 2967 . . 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 384   = wceq 1480  wcel 1987  wral 2908  cin 3559  wss 3560  𝒫 cpw 4136  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865
This theorem is referenced by:  ntrk1k3eqk13  37869
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