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Theorem smoeq 7334
 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 5246 . . . 4 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
31, 2feq12d 5946 . . 3 (𝐴 = 𝐵 → (𝐴:dom 𝐴⟶On ↔ 𝐵:dom 𝐵⟶On))
4 ordeq 5647 . . . 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 6102 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝑥) = (𝐵𝑥))
7 fveq1 6102 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝑦) = (𝐵𝑦))
86, 7eleq12d 2682 . . . . . 6 (𝐴 = 𝐵 → ((𝐴𝑥) ∈ (𝐴𝑦) ↔ (𝐵𝑥) ∈ (𝐵𝑦)))
98imbi2d 329 . . . . 5 (𝐴 = 𝐵 → ((𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) ↔ (𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1092ralbidv 2972 . . . 4 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) ↔ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
112raleqdv 3121 . . . . 5 (𝐴 = 𝐵 → (∀𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ ∀𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1211ralbidv 2969 . . . 4 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
132raleqdv 3121 . . . 4 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1410, 12, 133bitrd 293 . . 3 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) ↔ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
153, 5, 143anbi123d 1391 . 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 7330 . 2 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
17 df-smo 7330 . 2 (Smo 𝐵 ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1815, 16, 173bitr4g 302 1 (𝐴 = 𝐵 → (Smo 𝐴 ↔ Smo 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  dom cdm 5038  Ord word 5639  Oncon0 5640  ⟶wf 5800  ‘cfv 5804  Smo wsmo 7329 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-ord 5643  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-smo 7330 This theorem is referenced by:  smores3  7337  smo0  7342  cofsmo  8974  cfsmolem  8975  alephsing  8981
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