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Theorem smoel 8307
Description: If π‘₯ is less than 𝑦 then a strictly monotone function's value will be strictly less at π‘₯ than at 𝑦. (Contributed by Andrew Salmon, 22-Nov-2011.)
Assertion
Ref Expression
smoel ((Smo 𝐡 ∧ 𝐴 ∈ dom 𝐡 ∧ 𝐢 ∈ 𝐴) β†’ (π΅β€˜πΆ) ∈ (π΅β€˜π΄))

Proof of Theorem smoel
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smodm 8298 . . . . 5 (Smo 𝐡 β†’ Ord dom 𝐡)
2 ordtr1 6361 . . . . . . 7 (Ord dom 𝐡 β†’ ((𝐢 ∈ 𝐴 ∧ 𝐴 ∈ dom 𝐡) β†’ 𝐢 ∈ dom 𝐡))
32ancomsd 467 . . . . . 6 (Ord dom 𝐡 β†’ ((𝐴 ∈ dom 𝐡 ∧ 𝐢 ∈ 𝐴) β†’ 𝐢 ∈ dom 𝐡))
43expdimp 454 . . . . 5 ((Ord dom 𝐡 ∧ 𝐴 ∈ dom 𝐡) β†’ (𝐢 ∈ 𝐴 β†’ 𝐢 ∈ dom 𝐡))
51, 4sylan 581 . . . 4 ((Smo 𝐡 ∧ 𝐴 ∈ dom 𝐡) β†’ (𝐢 ∈ 𝐴 β†’ 𝐢 ∈ dom 𝐡))
6 df-smo 8293 . . . . . 6 (Smo 𝐡 ↔ (𝐡:dom 𝐡⟢On ∧ Ord dom 𝐡 ∧ βˆ€π‘₯ ∈ dom π΅βˆ€π‘¦ ∈ dom 𝐡(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦))))
7 eleq1 2822 . . . . . . . . . . 11 (π‘₯ = 𝐢 β†’ (π‘₯ ∈ 𝑦 ↔ 𝐢 ∈ 𝑦))
8 fveq2 6843 . . . . . . . . . . . 12 (π‘₯ = 𝐢 β†’ (π΅β€˜π‘₯) = (π΅β€˜πΆ))
98eleq1d 2819 . . . . . . . . . . 11 (π‘₯ = 𝐢 β†’ ((π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦) ↔ (π΅β€˜πΆ) ∈ (π΅β€˜π‘¦)))
107, 9imbi12d 345 . . . . . . . . . 10 (π‘₯ = 𝐢 β†’ ((π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦)) ↔ (𝐢 ∈ 𝑦 β†’ (π΅β€˜πΆ) ∈ (π΅β€˜π‘¦))))
11 eleq2 2823 . . . . . . . . . . 11 (𝑦 = 𝐴 β†’ (𝐢 ∈ 𝑦 ↔ 𝐢 ∈ 𝐴))
12 fveq2 6843 . . . . . . . . . . . 12 (𝑦 = 𝐴 β†’ (π΅β€˜π‘¦) = (π΅β€˜π΄))
1312eleq2d 2820 . . . . . . . . . . 11 (𝑦 = 𝐴 β†’ ((π΅β€˜πΆ) ∈ (π΅β€˜π‘¦) ↔ (π΅β€˜πΆ) ∈ (π΅β€˜π΄)))
1411, 13imbi12d 345 . . . . . . . . . 10 (𝑦 = 𝐴 β†’ ((𝐢 ∈ 𝑦 β†’ (π΅β€˜πΆ) ∈ (π΅β€˜π‘¦)) ↔ (𝐢 ∈ 𝐴 β†’ (π΅β€˜πΆ) ∈ (π΅β€˜π΄))))
1510, 14rspc2v 3589 . . . . . . . . 9 ((𝐢 ∈ dom 𝐡 ∧ 𝐴 ∈ dom 𝐡) β†’ (βˆ€π‘₯ ∈ dom π΅βˆ€π‘¦ ∈ dom 𝐡(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦)) β†’ (𝐢 ∈ 𝐴 β†’ (π΅β€˜πΆ) ∈ (π΅β€˜π΄))))
1615ancoms 460 . . . . . . . 8 ((𝐴 ∈ dom 𝐡 ∧ 𝐢 ∈ dom 𝐡) β†’ (βˆ€π‘₯ ∈ dom π΅βˆ€π‘¦ ∈ dom 𝐡(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦)) β†’ (𝐢 ∈ 𝐴 β†’ (π΅β€˜πΆ) ∈ (π΅β€˜π΄))))
1716com12 32 . . . . . . 7 (βˆ€π‘₯ ∈ dom π΅βˆ€π‘¦ ∈ dom 𝐡(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦)) β†’ ((𝐴 ∈ dom 𝐡 ∧ 𝐢 ∈ dom 𝐡) β†’ (𝐢 ∈ 𝐴 β†’ (π΅β€˜πΆ) ∈ (π΅β€˜π΄))))
18173ad2ant3 1136 . . . . . 6 ((𝐡:dom 𝐡⟢On ∧ Ord dom 𝐡 ∧ βˆ€π‘₯ ∈ dom π΅βˆ€π‘¦ ∈ dom 𝐡(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦))) β†’ ((𝐴 ∈ dom 𝐡 ∧ 𝐢 ∈ dom 𝐡) β†’ (𝐢 ∈ 𝐴 β†’ (π΅β€˜πΆ) ∈ (π΅β€˜π΄))))
196, 18sylbi 216 . . . . 5 (Smo 𝐡 β†’ ((𝐴 ∈ dom 𝐡 ∧ 𝐢 ∈ dom 𝐡) β†’ (𝐢 ∈ 𝐴 β†’ (π΅β€˜πΆ) ∈ (π΅β€˜π΄))))
2019expdimp 454 . . . 4 ((Smo 𝐡 ∧ 𝐴 ∈ dom 𝐡) β†’ (𝐢 ∈ dom 𝐡 β†’ (𝐢 ∈ 𝐴 β†’ (π΅β€˜πΆ) ∈ (π΅β€˜π΄))))
215, 20syld 47 . . 3 ((Smo 𝐡 ∧ 𝐴 ∈ dom 𝐡) β†’ (𝐢 ∈ 𝐴 β†’ (𝐢 ∈ 𝐴 β†’ (π΅β€˜πΆ) ∈ (π΅β€˜π΄))))
2221pm2.43d 53 . 2 ((Smo 𝐡 ∧ 𝐴 ∈ dom 𝐡) β†’ (𝐢 ∈ 𝐴 β†’ (π΅β€˜πΆ) ∈ (π΅β€˜π΄)))
23223impia 1118 1 ((Smo 𝐡 ∧ 𝐴 ∈ dom 𝐡 ∧ 𝐢 ∈ 𝐴) β†’ (π΅β€˜πΆ) ∈ (π΅β€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  dom cdm 5634  Ord word 6317  Oncon0 6318  βŸΆwf 6493  β€˜cfv 6497  Smo wsmo 8292
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 2704
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 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-tr 5224  df-ord 6321  df-iota 6449  df-fv 6505  df-smo 8293
This theorem is referenced by:  smoiun  8308  smoel2  8310
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