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Theorem smoel 6197
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 6188 . . . . 5 (Smo 𝐵 → Ord dom 𝐵)
2 ordtr1 4310 . . . . . . 7 (Ord dom 𝐵 → ((𝐶𝐴𝐴 ∈ dom 𝐵) → 𝐶 ∈ dom 𝐵))
32ancomsd 267 . . . . . 6 (Ord dom 𝐵 → ((𝐴 ∈ dom 𝐵𝐶𝐴) → 𝐶 ∈ dom 𝐵))
43expdimp 257 . . . . 5 ((Ord dom 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴𝐶 ∈ dom 𝐵))
51, 4sylan 281 . . . 4 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴𝐶 ∈ dom 𝐵))
6 df-smo 6183 . . . . . 6 (Smo 𝐵 ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
7 eleq1 2202 . . . . . . . . . . 11 (𝑥 = 𝐶 → (𝑥𝑦𝐶𝑦))
8 fveq2 5421 . . . . . . . . . . . 12 (𝑥 = 𝐶 → (𝐵𝑥) = (𝐵𝐶))
98eleq1d 2208 . . . . . . . . . . 11 (𝑥 = 𝐶 → ((𝐵𝑥) ∈ (𝐵𝑦) ↔ (𝐵𝐶) ∈ (𝐵𝑦)))
107, 9imbi12d 233 . . . . . . . . . 10 (𝑥 = 𝐶 → ((𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ (𝐶𝑦 → (𝐵𝐶) ∈ (𝐵𝑦))))
11 eleq2 2203 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝐶𝑦𝐶𝐴))
12 fveq2 5421 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝐵𝑦) = (𝐵𝐴))
1312eleq2d 2209 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((𝐵𝐶) ∈ (𝐵𝑦) ↔ (𝐵𝐶) ∈ (𝐵𝐴)))
1411, 13imbi12d 233 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝐶𝑦 → (𝐵𝐶) ∈ (𝐵𝑦)) ↔ (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
1510, 14rspc2v 2802 . . . . . . . . 9 ((𝐶 ∈ dom 𝐵𝐴 ∈ dom 𝐵) → (∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
1615ancoms 266 . . . . . . . 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 1004 . . . . . 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 120 . . . . 5 (Smo 𝐵 → ((𝐴 ∈ dom 𝐵𝐶 ∈ dom 𝐵) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
2019expdimp 257 . . . 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 1178 1 ((Smo 𝐵𝐴 ∈ dom 𝐵𝐶𝐴) → (𝐵𝐶) ∈ (𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 962   = wceq 1331  wcel 1480  wral 2416  Ord word 4284  Oncon0 4285  dom cdm 4539  wf 5119  cfv 5123  Smo wsmo 6182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-tr 4027  df-iord 4288  df-iota 5088  df-fv 5131  df-smo 6183
This theorem is referenced by:  smoiun  6198  smoel2  6200
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