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Theorem smoeq 7981
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 22 . . . 4 (𝐴 = 𝐵𝐴 = 𝐵)
2 dmeq 5770 . . . 4 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
31, 2feq12d 6498 . . 3 (𝐴 = 𝐵 → (𝐴:dom 𝐴⟶On ↔ 𝐵:dom 𝐵⟶On))
4 ordeq 6195 . . . 4 (dom 𝐴 = dom 𝐵 → (Ord dom 𝐴 ↔ Ord dom 𝐵))
52, 4syl 17 . . 3 (𝐴 = 𝐵 → (Ord dom 𝐴 ↔ Ord dom 𝐵))
6 fveq1 6665 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝑥) = (𝐵𝑥))
7 fveq1 6665 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝑦) = (𝐵𝑦))
86, 7eleq12d 2911 . . . . . 6 (𝐴 = 𝐵 → ((𝐴𝑥) ∈ (𝐴𝑦) ↔ (𝐵𝑥) ∈ (𝐵𝑦)))
98imbi2d 342 . . . . 5 (𝐴 = 𝐵 → ((𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) ↔ (𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1092ralbidv 3203 . . . 4 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) ↔ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
112raleqdv 3420 . . . . 5 (𝐴 = 𝐵 → (∀𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ ∀𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1211ralbidv 3201 . . . 4 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
132raleqdv 3420 . . . 4 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1410, 12, 133bitrd 306 . . 3 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) ↔ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
153, 5, 143anbi123d 1429 . 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 7977 . 2 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
17 df-smo 7977 . 2 (Smo 𝐵 ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1815, 16, 173bitr4g 315 1 (𝐴 = 𝐵 → (Smo 𝐴 ↔ Smo 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3a 1081   = wceq 1530  wcel 2107  wral 3142  dom cdm 5553  Ord word 6187  Oncon0 6188  wf 6347  cfv 6351  Smo wsmo 7976
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-tr 5169  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-ord 6191  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-smo 7977
This theorem is referenced by:  smores3  7984  smo0  7989  cofsmo  9683  cfsmolem  9684  alephsing  9690
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