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Theorem smoel 5915
Description: If 𝑥 is less than 𝑦 then a strictly monotone function's value will be strictly less at 𝑥 than at 𝑦. (Contributed by Andrew Salmon, 22-Nov-2011.)
Assertion
Ref Expression
smoel ((Smo 𝐵𝐴 ∈ dom 𝐵𝐶𝐴) → (𝐵𝐶) ∈ (𝐵𝐴))

Proof of Theorem smoel
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smodm 5906 . . . . 5 (Smo 𝐵 → Ord dom 𝐵)
2 ordtr1 4125 . . . . . . 7 (Ord dom 𝐵 → ((𝐶𝐴𝐴 ∈ dom 𝐵) → 𝐶 ∈ dom 𝐵))
32ancomsd 256 . . . . . 6 (Ord dom 𝐵 → ((𝐴 ∈ dom 𝐵𝐶𝐴) → 𝐶 ∈ dom 𝐵))
43expdimp 246 . . . . 5 ((Ord dom 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴𝐶 ∈ dom 𝐵))
51, 4sylan 267 . . . 4 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴𝐶 ∈ dom 𝐵))
6 df-smo 5901 . . . . . 6 (Smo 𝐵 ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
7 eleq1 2100 . . . . . . . . . . 11 (𝑥 = 𝐶 → (𝑥𝑦𝐶𝑦))
8 fveq2 5178 . . . . . . . . . . . 12 (𝑥 = 𝐶 → (𝐵𝑥) = (𝐵𝐶))
98eleq1d 2106 . . . . . . . . . . 11 (𝑥 = 𝐶 → ((𝐵𝑥) ∈ (𝐵𝑦) ↔ (𝐵𝐶) ∈ (𝐵𝑦)))
107, 9imbi12d 223 . . . . . . . . . 10 (𝑥 = 𝐶 → ((𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ (𝐶𝑦 → (𝐵𝐶) ∈ (𝐵𝑦))))
11 eleq2 2101 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝐶𝑦𝐶𝐴))
12 fveq2 5178 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝐵𝑦) = (𝐵𝐴))
1312eleq2d 2107 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((𝐵𝐶) ∈ (𝐵𝑦) ↔ (𝐵𝐶) ∈ (𝐵𝐴)))
1411, 13imbi12d 223 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝐶𝑦 → (𝐵𝐶) ∈ (𝐵𝑦)) ↔ (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
1510, 14rspc2v 2662 . . . . . . . . 9 ((𝐶 ∈ dom 𝐵𝐴 ∈ dom 𝐵) → (∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
1615ancoms 255 . . . . . . . 8 ((𝐴 ∈ dom 𝐵𝐶 ∈ dom 𝐵) → (∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
1716com12 27 . . . . . . 7 (∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) → ((𝐴 ∈ dom 𝐵𝐶 ∈ dom 𝐵) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
18173ad2ant3 927 . . . . . 6 ((𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))) → ((𝐴 ∈ dom 𝐵𝐶 ∈ dom 𝐵) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
196, 18sylbi 114 . . . . 5 (Smo 𝐵 → ((𝐴 ∈ dom 𝐵𝐶 ∈ dom 𝐵) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
2019expdimp 246 . . . 4 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐶 ∈ dom 𝐵 → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
215, 20syld 40 . . 3 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴 → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
2221pm2.43d 44 . 2 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴)))
23223impia 1101 1 ((Smo 𝐵𝐴 ∈ dom 𝐵𝐶𝐴) → (𝐵𝐶) ∈ (𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  w3a 885   = wceq 1243  wcel 1393  wral 2306  Ord word 4099  Oncon0 4100  dom cdm 4345  wf 4898  cfv 4902  Smo wsmo 5900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-tr 3855  df-iord 4103  df-iota 4867  df-fv 4910  df-smo 5901
This theorem is referenced by:  smoiun  5916  smoel2  5918
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