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Theorem smoeq 6117
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 4677 . . . 4 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
31, 2feq12d 5198 . . 3 (𝐴 = 𝐵 → (𝐴:dom 𝐴⟶On ↔ 𝐵:dom 𝐵⟶On))
4 ordeq 4232 . . . 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 5352 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝑥) = (𝐵𝑥))
7 fveq1 5352 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝑦) = (𝐵𝑦))
86, 7eleq12d 2170 . . . . . 6 (𝐴 = 𝐵 → ((𝐴𝑥) ∈ (𝐴𝑦) ↔ (𝐵𝑥) ∈ (𝐵𝑦)))
98imbi2d 229 . . . . 5 (𝐴 = 𝐵 → ((𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) ↔ (𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1092ralbidv 2418 . . . 4 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) ↔ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
112raleqdv 2590 . . . . 5 (𝐴 = 𝐵 → (∀𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ ∀𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1211ralbidv 2396 . . . 4 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
132raleqdv 2590 . . . 4 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1410, 12, 133bitrd 213 . . 3 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) ↔ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
153, 5, 143anbi123d 1258 . 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 6113 . 2 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
17 df-smo 6113 . 2 (Smo 𝐵 ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1815, 16, 173bitr4g 222 1 (𝐴 = 𝐵 → (Smo 𝐴 ↔ Smo 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  w3a 930   = wceq 1299  wcel 1448  wral 2375  Ord word 4222  Oncon0 4223  dom cdm 4477  wf 5055  cfv 5059  Smo wsmo 6112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-tr 3967  df-iord 4226  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-smo 6113
This theorem is referenced by:  smores3  6120  smo0  6125
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