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Theorem smoeq 8371
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 5906 . . . 4 (𝐴 = 𝐡 β†’ dom 𝐴 = dom 𝐡)
31, 2feq12d 6710 . . 3 (𝐴 = 𝐡 β†’ (𝐴:dom 𝐴⟢On ↔ 𝐡:dom 𝐡⟢On))
4 ordeq 6376 . . . 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 6896 . . . . . . 7 (𝐴 = 𝐡 β†’ (π΄β€˜π‘₯) = (π΅β€˜π‘₯))
7 fveq1 6896 . . . . . . 7 (𝐴 = 𝐡 β†’ (π΄β€˜π‘¦) = (π΅β€˜π‘¦))
86, 7eleq12d 2823 . . . . . 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 3215 . . . 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 3322 . . . . 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 3174 . . . 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 3322 . . . 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 1433 . 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 8367 . 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 8367 . 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 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058  dom cdm 5678  Ord word 6368  Oncon0 6369  βŸΆwf 6544  β€˜cfv 6548  Smo wsmo 8366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-ord 6372  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-smo 8367
This theorem is referenced by:  smores3  8374  smo0  8379  cofsmo  10293  cfsmolem  10294  alephsing  10300
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