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Theorem smoeq 8346
Description: Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
Assertion
Ref Expression
smoeq (𝐴 = 𝐡 β†’ (Smo 𝐴 ↔ Smo 𝐡))

Proof of Theorem smoeq
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 (𝐴 = 𝐡 β†’ 𝐴 = 𝐡)
2 dmeq 5894 . . . 4 (𝐴 = 𝐡 β†’ dom 𝐴 = dom 𝐡)
31, 2feq12d 6696 . . 3 (𝐴 = 𝐡 β†’ (𝐴:dom 𝐴⟢On ↔ 𝐡:dom 𝐡⟢On))
4 ordeq 6362 . . . 4 (dom 𝐴 = dom 𝐡 β†’ (Ord dom 𝐴 ↔ Ord dom 𝐡))
52, 4syl 17 . . 3 (𝐴 = 𝐡 β†’ (Ord dom 𝐴 ↔ Ord dom 𝐡))
6 fveq1 6881 . . . . . . 7 (𝐴 = 𝐡 β†’ (π΄β€˜π‘₯) = (π΅β€˜π‘₯))
7 fveq1 6881 . . . . . . 7 (𝐴 = 𝐡 β†’ (π΄β€˜π‘¦) = (π΅β€˜π‘¦))
86, 7eleq12d 2819 . . . . . 6 (𝐴 = 𝐡 β†’ ((π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦) ↔ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦)))
98imbi2d 340 . . . . 5 (𝐴 = 𝐡 β†’ ((π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)) ↔ (π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦))))
1092ralbidv 3210 . . . 4 (𝐴 = 𝐡 β†’ (βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦))))
112raleqdv 3317 . . . . 5 (𝐴 = 𝐡 β†’ (βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ dom 𝐡(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦))))
1211ralbidv 3169 . . . 4 (𝐴 = 𝐡 β†’ (βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐡(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦))))
132raleqdv 3317 . . . 4 (𝐴 = 𝐡 β†’ (βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐡(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ dom π΅βˆ€π‘¦ ∈ dom 𝐡(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦))))
1410, 12, 133bitrd 305 . . 3 (𝐴 = 𝐡 β†’ (βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ dom π΅βˆ€π‘¦ ∈ dom 𝐡(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦))))
153, 5, 143anbi123d 1432 . 2 (𝐴 = 𝐡 β†’ ((𝐴:dom 𝐴⟢On ∧ Ord dom 𝐴 ∧ βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦))) ↔ (𝐡:dom 𝐡⟢On ∧ Ord dom 𝐡 ∧ βˆ€π‘₯ ∈ dom π΅βˆ€π‘¦ ∈ dom 𝐡(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦)))))
16 df-smo 8342 . 2 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟢On ∧ Ord dom 𝐴 ∧ βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦))))
17 df-smo 8342 . 2 (Smo 𝐡 ↔ (𝐡:dom 𝐡⟢On ∧ Ord dom 𝐡 ∧ βˆ€π‘₯ ∈ dom π΅βˆ€π‘¦ ∈ dom 𝐡(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦))))
1815, 16, 173bitr4g 314 1 (𝐴 = 𝐡 β†’ (Smo 𝐴 ↔ Smo 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  dom cdm 5667  Ord word 6354  Oncon0 6355  βŸΆwf 6530  β€˜cfv 6534  Smo wsmo 8341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-ord 6358  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-smo 8342
This theorem is referenced by:  smores3  8349  smo0  8354  cofsmo  10261  cfsmolem  10262  alephsing  10268
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