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Theorem smoel 6103
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 6094 . . . . 5 (Smo 𝐵 → Ord dom 𝐵)
2 ordtr1 4239 . . . . . . 7 (Ord dom 𝐵 → ((𝐶𝐴𝐴 ∈ dom 𝐵) → 𝐶 ∈ dom 𝐵))
32ancomsd 266 . . . . . 6 (Ord dom 𝐵 → ((𝐴 ∈ dom 𝐵𝐶𝐴) → 𝐶 ∈ dom 𝐵))
43expdimp 256 . . . . 5 ((Ord dom 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴𝐶 ∈ dom 𝐵))
51, 4sylan 278 . . . 4 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐶𝐴𝐶 ∈ dom 𝐵))
6 df-smo 6089 . . . . . 6 (Smo 𝐵 ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
7 eleq1 2157 . . . . . . . . . . 11 (𝑥 = 𝐶 → (𝑥𝑦𝐶𝑦))
8 fveq2 5340 . . . . . . . . . . . 12 (𝑥 = 𝐶 → (𝐵𝑥) = (𝐵𝐶))
98eleq1d 2163 . . . . . . . . . . 11 (𝑥 = 𝐶 → ((𝐵𝑥) ∈ (𝐵𝑦) ↔ (𝐵𝐶) ∈ (𝐵𝑦)))
107, 9imbi12d 233 . . . . . . . . . 10 (𝑥 = 𝐶 → ((𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ (𝐶𝑦 → (𝐵𝐶) ∈ (𝐵𝑦))))
11 eleq2 2158 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝐶𝑦𝐶𝐴))
12 fveq2 5340 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝐵𝑦) = (𝐵𝐴))
1312eleq2d 2164 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((𝐵𝐶) ∈ (𝐵𝑦) ↔ (𝐵𝐶) ∈ (𝐵𝐴)))
1411, 13imbi12d 233 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝐶𝑦 → (𝐵𝐶) ∈ (𝐵𝑦)) ↔ (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
1510, 14rspc2v 2748 . . . . . . . . 9 ((𝐶 ∈ dom 𝐵𝐴 ∈ dom 𝐵) → (∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) → (𝐶𝐴 → (𝐵𝐶) ∈ (𝐵𝐴))))
1615ancoms 265 . . . . . . . 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 969 . . . . . 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 256 . . . 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 1143 1 ((Smo 𝐵𝐴 ∈ dom 𝐵𝐶𝐴) → (𝐵𝐶) ∈ (𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 927   = wceq 1296  wcel 1445  wral 2370  Ord word 4213  Oncon0 4214  dom cdm 4467  wf 5045  cfv 5049  Smo wsmo 6088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-tr 3959  df-iord 4217  df-iota 5014  df-fv 5057  df-smo 6089
This theorem is referenced by:  smoiun  6104  smoel2  6106
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