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Theorem smo11 7446
Description: A strictly monotone ordinal function is one-to-one. (Contributed by Mario Carneiro, 28-Feb-2013.)
Assertion
Ref Expression
smo11 ((𝐹:𝐴𝐵 ∧ Smo 𝐹) → 𝐹:𝐴1-1𝐵)

Proof of Theorem smo11
Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 473 . 2 ((𝐹:𝐴𝐵 ∧ Smo 𝐹) → 𝐹:𝐴𝐵)
2 ffn 6032 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 smodm2 7437 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)
4 ordelord 5733 . . . . . . . 8 ((Ord 𝐴𝑧𝐴) → Ord 𝑧)
54ex 450 . . . . . . 7 (Ord 𝐴 → (𝑧𝐴 → Ord 𝑧))
63, 5syl 17 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝑧𝐴 → Ord 𝑧))
7 ordelord 5733 . . . . . . . 8 ((Ord 𝐴𝑤𝐴) → Ord 𝑤)
87ex 450 . . . . . . 7 (Ord 𝐴 → (𝑤𝐴 → Ord 𝑤))
93, 8syl 17 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝑤𝐴 → Ord 𝑤))
106, 9anim12d 585 . . . . 5 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝑧𝐴𝑤𝐴) → (Ord 𝑧 ∧ Ord 𝑤)))
11 ordtri3or 5743 . . . . . . 7 ((Ord 𝑧 ∧ Ord 𝑤) → (𝑧𝑤𝑧 = 𝑤𝑤𝑧))
12 simp1rr 1125 . . . . . . . . . . 11 ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) ∧ 𝑧𝑤 ∧ (𝐹𝑧) = (𝐹𝑤)) → 𝑤𝐴)
13 smoel2 7445 . . . . . . . . . . . . . 14 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥𝐴𝑦𝑥)) → (𝐹𝑦) ∈ (𝐹𝑥))
1413ralrimivva 2968 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
1514adantr 481 . . . . . . . . . . . 12 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) → ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
16153ad2ant1 1080 . . . . . . . . . . 11 ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) ∧ 𝑧𝑤 ∧ (𝐹𝑧) = (𝐹𝑤)) → ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
17 simp2 1060 . . . . . . . . . . 11 ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) ∧ 𝑧𝑤 ∧ (𝐹𝑧) = (𝐹𝑤)) → 𝑧𝑤)
18 simp3 1061 . . . . . . . . . . 11 ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) ∧ 𝑧𝑤 ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝐹𝑧) = (𝐹𝑤))
19 fveq2 6178 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
2019eleq2d 2685 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → ((𝐹𝑦) ∈ (𝐹𝑥) ↔ (𝐹𝑦) ∈ (𝐹𝑤)))
2120raleqbi1dv 3141 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) ↔ ∀𝑦𝑤 (𝐹𝑦) ∈ (𝐹𝑤)))
2221rspcv 3300 . . . . . . . . . . . . . 14 (𝑤𝐴 → (∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) → ∀𝑦𝑤 (𝐹𝑦) ∈ (𝐹𝑤)))
23 fveq2 6178 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
2423eleq1d 2684 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → ((𝐹𝑦) ∈ (𝐹𝑤) ↔ (𝐹𝑧) ∈ (𝐹𝑤)))
2524rspccv 3301 . . . . . . . . . . . . . 14 (∀𝑦𝑤 (𝐹𝑦) ∈ (𝐹𝑤) → (𝑧𝑤 → (𝐹𝑧) ∈ (𝐹𝑤)))
2622, 25syl6 35 . . . . . . . . . . . . 13 (𝑤𝐴 → (∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) → (𝑧𝑤 → (𝐹𝑧) ∈ (𝐹𝑤))))
27263imp 1254 . . . . . . . . . . . 12 ((𝑤𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) ∧ 𝑧𝑤) → (𝐹𝑧) ∈ (𝐹𝑤))
28 eleq1 2687 . . . . . . . . . . . . 13 ((𝐹𝑧) = (𝐹𝑤) → ((𝐹𝑧) ∈ (𝐹𝑤) ↔ (𝐹𝑤) ∈ (𝐹𝑤)))
2928biimpac 503 . . . . . . . . . . . 12 (((𝐹𝑧) ∈ (𝐹𝑤) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝐹𝑤) ∈ (𝐹𝑤))
3027, 29sylan 488 . . . . . . . . . . 11 (((𝑤𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) ∧ 𝑧𝑤) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝐹𝑤) ∈ (𝐹𝑤))
3112, 16, 17, 18, 30syl31anc 1327 . . . . . . . . . 10 ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) ∧ 𝑧𝑤 ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝐹𝑤) ∈ (𝐹𝑤))
32 smofvon2 7438 . . . . . . . . . . . . 13 (Smo 𝐹 → (𝐹𝑤) ∈ On)
33 eloni 5721 . . . . . . . . . . . . 13 ((𝐹𝑤) ∈ On → Ord (𝐹𝑤))
34 ordirr 5729 . . . . . . . . . . . . 13 (Ord (𝐹𝑤) → ¬ (𝐹𝑤) ∈ (𝐹𝑤))
3532, 33, 343syl 18 . . . . . . . . . . . 12 (Smo 𝐹 → ¬ (𝐹𝑤) ∈ (𝐹𝑤))
3635ad2antlr 762 . . . . . . . . . . 11 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) → ¬ (𝐹𝑤) ∈ (𝐹𝑤))
37363ad2ant1 1080 . . . . . . . . . 10 ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) ∧ 𝑧𝑤 ∧ (𝐹𝑧) = (𝐹𝑤)) → ¬ (𝐹𝑤) ∈ (𝐹𝑤))
3831, 37pm2.21dd 186 . . . . . . . . 9 ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) ∧ 𝑧𝑤 ∧ (𝐹𝑧) = (𝐹𝑤)) → 𝑧 = 𝑤)
39383exp 1262 . . . . . . . 8 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) → (𝑧𝑤 → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
40 ax-1 6 . . . . . . . . 9 (𝑧 = 𝑤 → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
4140a1i 11 . . . . . . . 8 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) → (𝑧 = 𝑤 → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
42 simp1rl 1124 . . . . . . . . . . 11 ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) ∧ 𝑤𝑧 ∧ (𝐹𝑧) = (𝐹𝑤)) → 𝑧𝐴)
43153ad2ant1 1080 . . . . . . . . . . 11 ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) ∧ 𝑤𝑧 ∧ (𝐹𝑧) = (𝐹𝑤)) → ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
44 simp2 1060 . . . . . . . . . . 11 ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) ∧ 𝑤𝑧 ∧ (𝐹𝑧) = (𝐹𝑤)) → 𝑤𝑧)
45 simp3 1061 . . . . . . . . . . 11 ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) ∧ 𝑤𝑧 ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝐹𝑧) = (𝐹𝑤))
46 fveq2 6178 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
4746eleq2d 2685 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((𝐹𝑦) ∈ (𝐹𝑥) ↔ (𝐹𝑦) ∈ (𝐹𝑧)))
4847raleqbi1dv 3141 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) ↔ ∀𝑦𝑧 (𝐹𝑦) ∈ (𝐹𝑧)))
4948rspcv 3300 . . . . . . . . . . . . . 14 (𝑧𝐴 → (∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) → ∀𝑦𝑧 (𝐹𝑦) ∈ (𝐹𝑧)))
50 fveq2 6178 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
5150eleq1d 2684 . . . . . . . . . . . . . . 15 (𝑦 = 𝑤 → ((𝐹𝑦) ∈ (𝐹𝑧) ↔ (𝐹𝑤) ∈ (𝐹𝑧)))
5251rspccv 3301 . . . . . . . . . . . . . 14 (∀𝑦𝑧 (𝐹𝑦) ∈ (𝐹𝑧) → (𝑤𝑧 → (𝐹𝑤) ∈ (𝐹𝑧)))
5349, 52syl6 35 . . . . . . . . . . . . 13 (𝑧𝐴 → (∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) → (𝑤𝑧 → (𝐹𝑤) ∈ (𝐹𝑧))))
54533imp 1254 . . . . . . . . . . . 12 ((𝑧𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) ∧ 𝑤𝑧) → (𝐹𝑤) ∈ (𝐹𝑧))
55 eleq2 2688 . . . . . . . . . . . . 13 ((𝐹𝑧) = (𝐹𝑤) → ((𝐹𝑤) ∈ (𝐹𝑧) ↔ (𝐹𝑤) ∈ (𝐹𝑤)))
5655biimpac 503 . . . . . . . . . . . 12 (((𝐹𝑤) ∈ (𝐹𝑧) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝐹𝑤) ∈ (𝐹𝑤))
5754, 56sylan 488 . . . . . . . . . . 11 (((𝑧𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) ∧ 𝑤𝑧) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝐹𝑤) ∈ (𝐹𝑤))
5842, 43, 44, 45, 57syl31anc 1327 . . . . . . . . . 10 ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) ∧ 𝑤𝑧 ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝐹𝑤) ∈ (𝐹𝑤))
59363ad2ant1 1080 . . . . . . . . . 10 ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) ∧ 𝑤𝑧 ∧ (𝐹𝑧) = (𝐹𝑤)) → ¬ (𝐹𝑤) ∈ (𝐹𝑤))
6058, 59pm2.21dd 186 . . . . . . . . 9 ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) ∧ 𝑤𝑧 ∧ (𝐹𝑧) = (𝐹𝑤)) → 𝑧 = 𝑤)
61603exp 1262 . . . . . . . 8 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) → (𝑤𝑧 → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
6239, 41, 613jaod 1390 . . . . . . 7 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) → ((𝑧𝑤𝑧 = 𝑤𝑤𝑧) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
6311, 62syl5 34 . . . . . 6 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧𝐴𝑤𝐴)) → ((Ord 𝑧 ∧ Ord 𝑤) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
6463ex 450 . . . . 5 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝑧𝐴𝑤𝐴) → ((Ord 𝑧 ∧ Ord 𝑤) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))))
6510, 64mpdd 43 . . . 4 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝑧𝐴𝑤𝐴) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
6665ralrimivv 2967 . . 3 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ∀𝑧𝐴𝑤𝐴 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
672, 66sylan 488 . 2 ((𝐹:𝐴𝐵 ∧ Smo 𝐹) → ∀𝑧𝐴𝑤𝐴 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
68 dff13 6497 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑧𝐴𝑤𝐴 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
691, 67, 68sylanbrc 697 1 ((𝐹:𝐴𝐵 ∧ Smo 𝐹) → 𝐹:𝐴1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3o 1035  w3a 1036   = wceq 1481  wcel 1988  wral 2909  Ord word 5710  Oncon0 5711   Fn wfn 5871  wf 5872  1-1wf1 5873  cfv 5876  Smo wsmo 7427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-ord 5714  df-on 5715  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fv 5884  df-smo 7428
This theorem is referenced by:  smoiso2  7451  alephf1ALT  8911
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