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Theorem smoel 6319
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 6310 . . . . 5 (Smo 𝐵 → Ord dom 𝐵)
2 ordtr1 4403 . . . . . . 7 (Ord dom 𝐵 → ((𝐶𝐴𝐴 ∈ dom 𝐵) → 𝐶 ∈ dom 𝐵))
32ancomsd 269 . . . . . 6 (Ord dom 𝐵 → ((𝐴 ∈ dom 𝐵𝐶𝐴) → 𝐶 ∈ dom 𝐵))
43expdimp 259 . . . . 5 ((Ord dom 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴𝐶 ∈ dom 𝐵))
51, 4sylan 283 . . . 4 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴𝐶 ∈ dom 𝐵))
6 df-smo 6305 . . . . . 6 (Smo 𝐵 ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
7 eleq1 2252 . . . . . . . . . . 11 (𝑥 = 𝐶 → (𝑥𝑦𝐶𝑦))
8 fveq2 5530 . . . . . . . . . . . 12 (𝑥 = 𝐶 → (𝐵𝑥) = (𝐵𝐶))
98eleq1d 2258 . . . . . . . . . . 11 (𝑥 = 𝐶 → ((𝐵𝑥) ∈ (𝐵𝑦) ↔ (𝐵𝐶) ∈ (𝐵𝑦)))
107, 9imbi12d 234 . . . . . . . . . 10 (𝑥 = 𝐶 → ((𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ (𝐶𝑦 → (𝐵𝐶) ∈ (𝐵𝑦))))
11 eleq2 2253 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝐶𝑦𝐶𝐴))
12 fveq2 5530 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝐵𝑦) = (𝐵𝐴))
1312eleq2d 2259 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((𝐵𝐶) ∈ (𝐵𝑦) ↔ (𝐵𝐶) ∈ (𝐵𝐴)))
1411, 13imbi12d 234 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝐶𝑦 → (𝐵𝐶) ∈ (𝐵𝑦)) ↔ (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
1510, 14rspc2v 2869 . . . . . . . . 9 ((𝐶 ∈ dom 𝐵𝐴 ∈ dom 𝐵) → (∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
1615ancoms 268 . . . . . . . 8 ((𝐴 ∈ dom 𝐵𝐶 ∈ dom 𝐵) → (∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
1716com12 30 . . . . . . 7 (∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) → ((𝐴 ∈ dom 𝐵𝐶 ∈ dom 𝐵) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
18173ad2ant3 1022 . . . . . 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 121 . . . . 5 (Smo 𝐵 → ((𝐴 ∈ dom 𝐵𝐶 ∈ dom 𝐵) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
2019expdimp 259 . . . 4 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐶 ∈ dom 𝐵 → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
215, 20syld 45 . . 3 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴 → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
2221pm2.43d 50 . 2 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴)))
23223impia 1202 1 ((Smo 𝐵𝐴 ∈ dom 𝐵𝐶𝐴) → (𝐵𝐶) ∈ (𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wcel 2160  wral 2468  Ord word 4377  Oncon0 4378  dom cdm 4641  wf 5227  cfv 5231  Smo wsmo 6304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-tr 4117  df-iord 4381  df-iota 5193  df-fv 5239  df-smo 6305
This theorem is referenced by:  smoiun  6320  smoel2  6322
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