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Theorem smoeq 5905
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 4535 . . . 4 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
31, 2feq12d 5036 . . 3 (𝐴 = 𝐵 → (𝐴:dom 𝐴⟶On ↔ 𝐵:dom 𝐵⟶On))
4 ordeq 4109 . . . 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 5177 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝑥) = (𝐵𝑥))
7 fveq1 5177 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝑦) = (𝐵𝑦))
86, 7eleq12d 2108 . . . . . 6 (𝐴 = 𝐵 → ((𝐴𝑥) ∈ (𝐴𝑦) ↔ (𝐵𝑥) ∈ (𝐵𝑦)))
98imbi2d 219 . . . . 5 (𝐴 = 𝐵 → ((𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) ↔ (𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1092ralbidv 2348 . . . 4 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) ↔ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
112raleqdv 2511 . . . . 5 (𝐴 = 𝐵 → (∀𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ ∀𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1211ralbidv 2326 . . . 4 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
132raleqdv 2511 . . . 4 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦)) ↔ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
1410, 12, 133bitrd 203 . . 3 (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) ↔ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
153, 5, 143anbi123d 1207 . 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 5901 . 2 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
17 df-smo 5901 . 2 (Smo 𝐵 ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (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 885   = wceq 1243  wcel 1393  wral 2306  Ord word 4099  Oncon0 4100  dom cdm 4345  wf 4898  cfv 4902  Smo wsmo 5900
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-tr 3855  df-iord 4103  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-smo 5901
This theorem is referenced by:  smores3  5908  smo0  5913
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