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Theorem smoeq 5939
Description: Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
Assertion
Ref Expression
smoeq (𝐴 = 𝐵 → (Smo 𝐴 ↔ Smo 𝐵))

Proof of Theorem smoeq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . 4 (𝐴 = 𝐵𝐴 = 𝐵)
2 dmeq 4563 . . . 4 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
31, 2feq12d 5067 . . 3 (𝐴 = 𝐵 → (𝐴:dom 𝐴⟶On ↔ 𝐵:dom 𝐵⟶On))
4 ordeq 4135 . . . 4 (dom 𝐴 = dom 𝐵 → (Ord dom 𝐴 ↔ Ord dom 𝐵))
52, 4syl 14 . . 3 (𝐴 = 𝐵 → (Ord dom 𝐴 ↔ Ord dom 𝐵))
6 fveq1 5208 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝑥) = (𝐵𝑥))
7 fveq1 5208 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝑦) = (𝐵𝑦))
86, 7eleq12d 2150 . . . . . 6 (𝐴 = 𝐵 → ((𝐴𝑥) ∈ (𝐴𝑦) ↔ (𝐵𝑥) ∈ (𝐵𝑦)))
98imbi2d 228 . . . . 5 (𝐴 = 𝐵 → ((𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) ↔ (𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1092ralbidv 2391 . . . 4 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) ↔ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
112raleqdv 2556 . . . . 5 (𝐴 = 𝐵 → (∀𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ ∀𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1211ralbidv 2369 . . . 4 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
132raleqdv 2556 . . . 4 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1410, 12, 133bitrd 212 . . 3 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) ↔ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
153, 5, 143anbi123d 1244 . 2 (𝐴 = 𝐵 → ((𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))) ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)))))
16 df-smo 5935 . 2 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
17 df-smo 5935 . 2 (Smo 𝐵 ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1815, 16, 173bitr4g 221 1 (𝐴 = 𝐵 → (Smo 𝐴 ↔ Smo 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  w3a 920   = wceq 1285  wcel 1434  wral 2349  Ord word 4125  Oncon0 4126  dom cdm 4371  wf 4928  cfv 4932  Smo wsmo 5934
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 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-tr 3884  df-iord 4129  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940  df-smo 5935
This theorem is referenced by:  smores3  5942  smo0  5947
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