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Theorem smoiso 7730
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 𝐴𝐵 ⊆ On) → Smo 𝐹)

Proof of Theorem smoiso
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isof1o 6833 . . . 4 (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
2 f1of 6382 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
31, 2syl 17 . . 3 (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴𝐵)
4 ffdm 6303 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
54simpld 490 . . . . 5 (𝐹:𝐴𝐵𝐹:dom 𝐹𝐵)
6 fss 6295 . . . . 5 ((𝐹:dom 𝐹𝐵𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
75, 6sylan 575 . . . 4 ((𝐹:𝐴𝐵𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
873adant2 1165 . . 3 ((𝐹:𝐴𝐵 ∧ Ord 𝐴𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
93, 8syl3an1 1206 . 2 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
10 fdm 6290 . . . . . 6 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
1110eqcomd 2831 . . . . 5 (𝐹:𝐴𝐵𝐴 = dom 𝐹)
12 ordeq 5974 . . . . 5 (𝐴 = dom 𝐹 → (Ord 𝐴 ↔ Ord dom 𝐹))
131, 2, 11, 124syl 19 . . . 4 (𝐹 Isom E , E (𝐴, 𝐵) → (Ord 𝐴 ↔ Ord dom 𝐹))
1413biimpa 470 . . 3 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴) → Ord dom 𝐹)
15143adant3 1166 . 2 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → Ord dom 𝐹)
1610eleq2d 2892 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑥 ∈ dom 𝐹𝑥𝐴))
1710eleq2d 2892 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑦 ∈ dom 𝐹𝑦𝐴))
1816, 17anbi12d 624 . . . . . 6 (𝐹:𝐴𝐵 → ((𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹) ↔ (𝑥𝐴𝑦𝐴)))
191, 2, 183syl 18 . . . . 5 (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹) ↔ (𝑥𝐴𝑦𝐴)))
20 isorel 6836 . . . . . . . 8 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦 ↔ (𝐹𝑥) E (𝐹𝑦)))
21 epel 5260 . . . . . . . 8 (𝑥 E 𝑦𝑥𝑦)
22 fvex 6450 . . . . . . . . 9 (𝐹𝑦) ∈ V
2322epeli 5259 . . . . . . . 8 ((𝐹𝑥) E (𝐹𝑦) ↔ (𝐹𝑥) ∈ (𝐹𝑦))
2420, 21, 233bitr3g 305 . . . . . . 7 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 ↔ (𝐹𝑥) ∈ (𝐹𝑦)))
2524biimpd 221 . . . . . 6 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦)))
2625ex 403 . . . . 5 (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥𝐴𝑦𝐴) → (𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦))))
2719, 26sylbid 232 . . . 4 (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹) → (𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦))))
2827ralrimivv 3179 . . 3 (𝐹 Isom E , E (𝐴, 𝐵) → ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦)))
29283ad2ant1 1167 . 2 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦)))
30 df-smo 7714 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑥) ∈ (𝐹𝑦))))
319, 15, 29, 30syl3anbrc 1447 1 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → Smo 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wral 3117  wss 3798   class class class wbr 4875   E cep 5256  dom cdm 5346  Ord word 5966  Oncon0 5967  wf 6123  1-1-ontowf1o 6126  cfv 6127   Isom wiso 6128  Smo wsmo 7713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-tr 4978  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-ord 5970  df-iota 6090  df-fn 6130  df-f 6131  df-f1 6132  df-f1o 6134  df-fv 6135  df-isom 6136  df-smo 7714
This theorem is referenced by:  smoiso2  7737
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