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Theorem smoiso 5835
Description: If 𝐹 is an isomorphism from an ordinal A onto B, which is a subset of the ordinals, then 𝐹 is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)
Assertion
Ref Expression
smoiso ((𝐹 Isom E , E (A, B) Ord A B ⊆ On) → Smo 𝐹)

Proof of Theorem smoiso
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isof1o 5368 . . . 4 (𝐹 Isom E , E (A, B) → 𝐹:A1-1-ontoB)
2 f1of 5047 . . . 4 (𝐹:A1-1-ontoB𝐹:AB)
31, 2syl 14 . . 3 (𝐹 Isom E , E (A, B) → 𝐹:AB)
4 ffdm 4982 . . . . . 6 (𝐹:AB → (𝐹:dom 𝐹B dom 𝐹A))
54simpld 105 . . . . 5 (𝐹:AB𝐹:dom 𝐹B)
6 fss 4976 . . . . 5 ((𝐹:dom 𝐹B B ⊆ On) → 𝐹:dom 𝐹⟶On)
75, 6sylan 267 . . . 4 ((𝐹:AB B ⊆ On) → 𝐹:dom 𝐹⟶On)
873adant2 909 . . 3 ((𝐹:AB Ord A B ⊆ On) → 𝐹:dom 𝐹⟶On)
93, 8syl3an1 1152 . 2 ((𝐹 Isom E , E (A, B) Ord A B ⊆ On) → 𝐹:dom 𝐹⟶On)
10 fdm 4972 . . . . . 6 (𝐹:AB → dom 𝐹 = A)
1110eqcomd 2023 . . . . 5 (𝐹:ABA = dom 𝐹)
12 ordeq 4054 . . . . 5 (A = dom 𝐹 → (Ord A ↔ Ord dom 𝐹))
131, 2, 11, 124syl 18 . . . 4 (𝐹 Isom E , E (A, B) → (Ord A ↔ Ord dom 𝐹))
1413biimpa 280 . . 3 ((𝐹 Isom E , E (A, B) Ord A) → Ord dom 𝐹)
15143adant3 910 . 2 ((𝐹 Isom E , E (A, B) Ord A B ⊆ On) → Ord dom 𝐹)
1610eleq2d 2085 . . . . . . 7 (𝐹:AB → (x dom 𝐹x A))
1710eleq2d 2085 . . . . . . 7 (𝐹:AB → (y dom 𝐹y A))
1816, 17anbi12d 445 . . . . . 6 (𝐹:AB → ((x dom 𝐹 y dom 𝐹) ↔ (x A y A)))
191, 2, 183syl 17 . . . . 5 (𝐹 Isom E , E (A, B) → ((x dom 𝐹 y dom 𝐹) ↔ (x A y A)))
20 epel 3999 . . . . . . . . 9 (x E yx y)
21 isorel 5369 . . . . . . . . 9 ((𝐹 Isom E , E (A, B) (x A y A)) → (x E y ↔ (𝐹x) E (𝐹y)))
2220, 21syl5bbr 183 . . . . . . . 8 ((𝐹 Isom E , E (A, B) (x A y A)) → (x y ↔ (𝐹x) E (𝐹y)))
23 ffn 4968 . . . . . . . . . . 11 (𝐹:AB𝐹 Fn A)
243, 23syl 14 . . . . . . . . . 10 (𝐹 Isom E , E (A, B) → 𝐹 Fn A)
2524adantr 261 . . . . . . . . 9 ((𝐹 Isom E , E (A, B) (x A y A)) → 𝐹 Fn A)
26 simprr 472 . . . . . . . . 9 ((𝐹 Isom E , E (A, B) (x A y A)) → y A)
27 funfvex 5113 . . . . . . . . . . 11 ((Fun 𝐹 y dom 𝐹) → (𝐹y) V)
2827funfni 4921 . . . . . . . . . 10 ((𝐹 Fn A y A) → (𝐹y) V)
29 epelg 3997 . . . . . . . . . 10 ((𝐹y) V → ((𝐹x) E (𝐹y) ↔ (𝐹x) (𝐹y)))
3028, 29syl 14 . . . . . . . . 9 ((𝐹 Fn A y A) → ((𝐹x) E (𝐹y) ↔ (𝐹x) (𝐹y)))
3125, 26, 30syl2anc 393 . . . . . . . 8 ((𝐹 Isom E , E (A, B) (x A y A)) → ((𝐹x) E (𝐹y) ↔ (𝐹x) (𝐹y)))
3222, 31bitrd 177 . . . . . . 7 ((𝐹 Isom E , E (A, B) (x A y A)) → (x y ↔ (𝐹x) (𝐹y)))
3332biimpd 132 . . . . . 6 ((𝐹 Isom E , E (A, B) (x A y A)) → (x y → (𝐹x) (𝐹y)))
3433ex 108 . . . . 5 (𝐹 Isom E , E (A, B) → ((x A y A) → (x y → (𝐹x) (𝐹y))))
3519, 34sylbid 139 . . . 4 (𝐹 Isom E , E (A, B) → ((x dom 𝐹 y dom 𝐹) → (x y → (𝐹x) (𝐹y))))
3635ralrimivv 2374 . . 3 (𝐹 Isom E , E (A, B) → x dom 𝐹y dom 𝐹(x y → (𝐹x) (𝐹y)))
37363ad2ant1 911 . 2 ((𝐹 Isom E , E (A, B) Ord A B ⊆ On) → x dom 𝐹y dom 𝐹(x y → (𝐹x) (𝐹y)))
38 df-smo 5819 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On Ord dom 𝐹 x dom 𝐹y dom 𝐹(x y → (𝐹x) (𝐹y))))
399, 15, 37, 38syl3anbrc 1073 1 ((𝐹 Isom E , E (A, B) Ord A B ⊆ On) → Smo 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 871   = wceq 1226   wcel 1370  wral 2280  Vcvv 2531  wss 2890   class class class wbr 3734   E cep 3994  Ord word 4044  Oncon0 4045  dom cdm 4268   Fn wfn 4820  wf 4821  1-1-ontowf1o 4824  cfv 4825   Isom wiso 4826  Smo wsmo 5818
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-tr 3825  df-eprel 3996  df-id 4000  df-iord 4048  df-cnv 4276  df-co 4277  df-dm 4278  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-f1o 4832  df-fv 4833  df-isom 4834  df-smo 5819
This theorem is referenced by: (None)
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