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Theorem smoeq 8297
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 5860 . . . 4 (𝐴 = 𝐡 β†’ dom 𝐴 = dom 𝐡)
31, 2feq12d 6657 . . 3 (𝐴 = 𝐡 β†’ (𝐴:dom 𝐴⟢On ↔ 𝐡:dom 𝐡⟢On))
4 ordeq 6325 . . . 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 6842 . . . . . . 7 (𝐴 = 𝐡 β†’ (π΄β€˜π‘₯) = (π΅β€˜π‘₯))
7 fveq1 6842 . . . . . . 7 (𝐴 = 𝐡 β†’ (π΄β€˜π‘¦) = (π΅β€˜π‘¦))
86, 7eleq12d 2828 . . . . . 6 (𝐴 = 𝐡 β†’ ((π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦) ↔ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦)))
98imbi2d 341 . . . . 5 (𝐴 = 𝐡 β†’ ((π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)) ↔ (π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦))))
1092ralbidv 3209 . . . 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 3312 . . . . 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 3171 . . . 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 3312 . . . 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 1437 . 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 8293 . 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 8293 . 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 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  dom cdm 5634  Ord word 6317  Oncon0 6318  βŸΆwf 6493  β€˜cfv 6497  Smo wsmo 8292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-ord 6321  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-smo 8293
This theorem is referenced by:  smores3  8300  smo0  8305  cofsmo  10210  cfsmolem  10211  alephsing  10217
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