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Theorem smoeq 5846
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 4478 . . . 4 (A = B → dom A = dom B)
31, 2feq12d 4979 . . 3 (A = B → (A:dom A⟶On ↔ B:dom B⟶On))
4 ordeq 4075 . . . 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 5120 . . . . . . 7 (A = B → (Ax) = (Bx))
7 fveq1 5120 . . . . . . 7 (A = B → (Ay) = (By))
86, 7eleq12d 2105 . . . . . 6 (A = B → ((Ax) (Ay) ↔ (Bx) (By)))
98imbi2d 219 . . . . 5 (A = B → ((x y → (Ax) (Ay)) ↔ (x y → (Bx) (By))))
1092ralbidv 2342 . . . 4 (A = B → (x dom Ay dom A(x y → (Ax) (Ay)) ↔ x dom Ay dom A(x y → (Bx) (By))))
112raleqdv 2505 . . . . 5 (A = B → (y dom A(x y → (Bx) (By)) ↔ y dom B(x y → (Bx) (By))))
1211ralbidv 2320 . . . 4 (A = B → (x dom Ay dom A(x y → (Bx) (By)) ↔ x dom Ay dom B(x y → (Bx) (By))))
132raleqdv 2505 . . . 4 (A = B → (x dom Ay dom B(x y → (Bx) (By)) ↔ x dom By dom B(x y → (Bx) (By))))
1410, 12, 133bitrd 203 . . 3 (A = B → (x dom Ay dom A(x y → (Ax) (Ay)) ↔ x dom By dom B(x y → (Bx) (By))))
153, 5, 143anbi123d 1206 . 2 (A = B → ((A:dom A⟶On Ord dom A x dom Ay dom A(x y → (Ax) (Ay))) ↔ (B:dom B⟶On Ord dom B x dom By dom B(x y → (Bx) (By)))))
16 df-smo 5842 . 2 (Smo A ↔ (A:dom A⟶On Ord dom A x dom Ay dom A(x y → (Ax) (Ay))))
17 df-smo 5842 . 2 (Smo B ↔ (B:dom B⟶On Ord dom B x dom By dom B(x y → (Bx) (By))))
1815, 16, 173bitr4g 212 1 (A = B → (Smo A ↔ Smo B))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   w3a 884   = wceq 1242   wcel 1390  wral 2300  Ord word 4065  Oncon0 4066  dom cdm 4288  wf 4841  cfv 4845  Smo wsmo 5841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-tr 3846  df-iord 4069  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-smo 5842
This theorem is referenced by:  smores3  5849  smo0  5854
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