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Theorem smoiso 8309
Description: If 𝐹 is an isomorphism from an ordinal 𝐴 onto 𝐡, 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, 𝐡) ∧ Ord 𝐴 ∧ 𝐡 βŠ† On) β†’ Smo 𝐹)

Proof of Theorem smoiso
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isof1o 7269 . . . 4 (𝐹 Isom E , E (𝐴, 𝐡) β†’ 𝐹:𝐴–1-1-onto→𝐡)
2 f1of 6785 . . . 4 (𝐹:𝐴–1-1-onto→𝐡 β†’ 𝐹:𝐴⟢𝐡)
31, 2syl 17 . . 3 (𝐹 Isom E , E (𝐴, 𝐡) β†’ 𝐹:𝐴⟢𝐡)
4 ffdm 6699 . . . . . 6 (𝐹:𝐴⟢𝐡 β†’ (𝐹:dom 𝐹⟢𝐡 ∧ dom 𝐹 βŠ† 𝐴))
54simpld 496 . . . . 5 (𝐹:𝐴⟢𝐡 β†’ 𝐹:dom 𝐹⟢𝐡)
6 fss 6686 . . . . 5 ((𝐹:dom 𝐹⟢𝐡 ∧ 𝐡 βŠ† On) β†’ 𝐹:dom 𝐹⟢On)
75, 6sylan 581 . . . 4 ((𝐹:𝐴⟢𝐡 ∧ 𝐡 βŠ† On) β†’ 𝐹:dom 𝐹⟢On)
873adant2 1132 . . 3 ((𝐹:𝐴⟢𝐡 ∧ Ord 𝐴 ∧ 𝐡 βŠ† On) β†’ 𝐹:dom 𝐹⟢On)
93, 8syl3an1 1164 . 2 ((𝐹 Isom E , E (𝐴, 𝐡) ∧ Ord 𝐴 ∧ 𝐡 βŠ† On) β†’ 𝐹:dom 𝐹⟢On)
10 fdm 6678 . . . . . 6 (𝐹:𝐴⟢𝐡 β†’ dom 𝐹 = 𝐴)
1110eqcomd 2739 . . . . 5 (𝐹:𝐴⟢𝐡 β†’ 𝐴 = dom 𝐹)
12 ordeq 6325 . . . . 5 (𝐴 = dom 𝐹 β†’ (Ord 𝐴 ↔ Ord dom 𝐹))
131, 2, 11, 124syl 19 . . . 4 (𝐹 Isom E , E (𝐴, 𝐡) β†’ (Ord 𝐴 ↔ Ord dom 𝐹))
1413biimpa 478 . . 3 ((𝐹 Isom E , E (𝐴, 𝐡) ∧ Ord 𝐴) β†’ Ord dom 𝐹)
15143adant3 1133 . 2 ((𝐹 Isom E , E (𝐴, 𝐡) ∧ Ord 𝐴 ∧ 𝐡 βŠ† On) β†’ Ord dom 𝐹)
1610eleq2d 2820 . . . . . . 7 (𝐹:𝐴⟢𝐡 β†’ (π‘₯ ∈ dom 𝐹 ↔ π‘₯ ∈ 𝐴))
1710eleq2d 2820 . . . . . . 7 (𝐹:𝐴⟢𝐡 β†’ (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ 𝐴))
1816, 17anbi12d 632 . . . . . 6 (𝐹:𝐴⟢𝐡 β†’ ((π‘₯ ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) ↔ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
191, 2, 183syl 18 . . . . 5 (𝐹 Isom E , E (𝐴, 𝐡) β†’ ((π‘₯ ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) ↔ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
20 isorel 7272 . . . . . . . 8 ((𝐹 Isom E , E (𝐴, 𝐡) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (π‘₯ E 𝑦 ↔ (πΉβ€˜π‘₯) E (πΉβ€˜π‘¦)))
21 epel 5541 . . . . . . . 8 (π‘₯ E 𝑦 ↔ π‘₯ ∈ 𝑦)
22 fvex 6856 . . . . . . . . 9 (πΉβ€˜π‘¦) ∈ V
2322epeli 5540 . . . . . . . 8 ((πΉβ€˜π‘₯) E (πΉβ€˜π‘¦) ↔ (πΉβ€˜π‘₯) ∈ (πΉβ€˜π‘¦))
2420, 21, 233bitr3g 313 . . . . . . 7 ((𝐹 Isom E , E (𝐴, 𝐡) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (π‘₯ ∈ 𝑦 ↔ (πΉβ€˜π‘₯) ∈ (πΉβ€˜π‘¦)))
2524biimpd 228 . . . . . 6 ((𝐹 Isom E , E (𝐴, 𝐡) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (π‘₯ ∈ 𝑦 β†’ (πΉβ€˜π‘₯) ∈ (πΉβ€˜π‘¦)))
2625ex 414 . . . . 5 (𝐹 Isom E , E (𝐴, 𝐡) β†’ ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ ∈ 𝑦 β†’ (πΉβ€˜π‘₯) ∈ (πΉβ€˜π‘¦))))
2719, 26sylbid 239 . . . 4 (𝐹 Isom E , E (𝐴, 𝐡) β†’ ((π‘₯ ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) β†’ (π‘₯ ∈ 𝑦 β†’ (πΉβ€˜π‘₯) ∈ (πΉβ€˜π‘¦))))
2827ralrimivv 3192 . . 3 (𝐹 Isom E , E (𝐴, 𝐡) β†’ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ dom 𝐹(π‘₯ ∈ 𝑦 β†’ (πΉβ€˜π‘₯) ∈ (πΉβ€˜π‘¦)))
29283ad2ant1 1134 . 2 ((𝐹 Isom E , E (𝐴, 𝐡) ∧ Ord 𝐴 ∧ 𝐡 βŠ† On) β†’ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ dom 𝐹(π‘₯ ∈ 𝑦 β†’ (πΉβ€˜π‘₯) ∈ (πΉβ€˜π‘¦)))
30 df-smo 8293 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟢On ∧ Ord dom 𝐹 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ dom 𝐹(π‘₯ ∈ 𝑦 β†’ (πΉβ€˜π‘₯) ∈ (πΉβ€˜π‘¦))))
319, 15, 29, 30syl3anbrc 1344 1 ((𝐹 Isom E , E (𝐴, 𝐡) ∧ Ord 𝐴 ∧ 𝐡 βŠ† On) β†’ Smo 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   βŠ† wss 3911   class class class wbr 5106   E cep 5537  dom cdm 5634  Ord word 6317  Oncon0 6318  βŸΆwf 6493  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497   Isom wiso 6498  Smo wsmo 8292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-ord 6321  df-iota 6449  df-fn 6500  df-f 6501  df-f1 6502  df-f1o 6504  df-fv 6505  df-isom 6506  df-smo 8293
This theorem is referenced by:  smoiso2  8316
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