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Theorem smoel 5969
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 5960 . . . . 5 (Smo 𝐵 → Ord dom 𝐵)
2 ordtr1 4171 . . . . . . 7 (Ord dom 𝐵 → ((𝐶𝐴𝐴 ∈ dom 𝐵) → 𝐶 ∈ dom 𝐵))
32ancomsd 265 . . . . . 6 (Ord dom 𝐵 → ((𝐴 ∈ dom 𝐵𝐶𝐴) → 𝐶 ∈ dom 𝐵))
43expdimp 255 . . . . 5 ((Ord dom 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴𝐶 ∈ dom 𝐵))
51, 4sylan 277 . . . 4 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴𝐶 ∈ dom 𝐵))
6 df-smo 5955 . . . . . 6 (Smo 𝐵 ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
7 eleq1 2145 . . . . . . . . . . 11 (𝑥 = 𝐶 → (𝑥𝑦𝐶𝑦))
8 fveq2 5229 . . . . . . . . . . . 12 (𝑥 = 𝐶 → (𝐵𝑥) = (𝐵𝐶))
98eleq1d 2151 . . . . . . . . . . 11 (𝑥 = 𝐶 → ((𝐵𝑥) ∈ (𝐵𝑦) ↔ (𝐵𝐶) ∈ (𝐵𝑦)))
107, 9imbi12d 232 . . . . . . . . . 10 (𝑥 = 𝐶 → ((𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ (𝐶𝑦 → (𝐵𝐶) ∈ (𝐵𝑦))))
11 eleq2 2146 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝐶𝑦𝐶𝐴))
12 fveq2 5229 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝐵𝑦) = (𝐵𝐴))
1312eleq2d 2152 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((𝐵𝐶) ∈ (𝐵𝑦) ↔ (𝐵𝐶) ∈ (𝐵𝐴)))
1411, 13imbi12d 232 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝐶𝑦 → (𝐵𝐶) ∈ (𝐵𝑦)) ↔ (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
1510, 14rspc2v 2721 . . . . . . . . 9 ((𝐶 ∈ dom 𝐵𝐴 ∈ dom 𝐵) → (∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
1615ancoms 264 . . . . . . . 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 962 . . . . . 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 119 . . . . 5 (Smo 𝐵 → ((𝐴 ∈ dom 𝐵𝐶 ∈ dom 𝐵) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
2019expdimp 255 . . . 4 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐶 ∈ dom 𝐵 → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
215, 20syld 44 . . 3 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴 → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
2221pm2.43d 49 . 2 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴)))
23223impia 1136 1 ((Smo 𝐵𝐴 ∈ dom 𝐵𝐶𝐴) → (𝐵𝐶) ∈ (𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wcel 1434  wral 2353  Ord word 4145  Oncon0 4146  dom cdm 4391  wf 4948  cfv 4952  Smo wsmo 5954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-tr 3896  df-iord 4149  df-iota 4917  df-fv 4960  df-smo 5955
This theorem is referenced by:  smoiun  5970  smoel2  5972
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