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Theorem r111 8585
Description: The cumulative hierarchy is a one-to-one function. (Contributed by Mario Carneiro, 19-Apr-2013.)
Assertion
Ref Expression
r111 𝑅1:On–1-1→V

Proof of Theorem r111
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1fnon 8577 . . 3 𝑅1 Fn On
2 dffn2 6006 . . 3 (𝑅1 Fn On ↔ 𝑅1:On⟶V)
31, 2mpbi 220 . 2 𝑅1:On⟶V
4 eloni 5694 . . . . 5 (𝑥 ∈ On → Ord 𝑥)
5 eloni 5694 . . . . 5 (𝑦 ∈ On → Ord 𝑦)
6 ordtri3or 5716 . . . . 5 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
74, 5, 6syl2an 494 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
8 sdomirr 8044 . . . . . . . . 9 ¬ (𝑅1𝑦) ≺ (𝑅1𝑦)
9 r1sdom 8584 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝑥𝑦) → (𝑅1𝑥) ≺ (𝑅1𝑦))
10 breq1 4618 . . . . . . . . . 10 ((𝑅1𝑥) = (𝑅1𝑦) → ((𝑅1𝑥) ≺ (𝑅1𝑦) ↔ (𝑅1𝑦) ≺ (𝑅1𝑦)))
119, 10syl5ibcom 235 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝑥𝑦) → ((𝑅1𝑥) = (𝑅1𝑦) → (𝑅1𝑦) ≺ (𝑅1𝑦)))
128, 11mtoi 190 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝑥𝑦) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
13123adant1 1077 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
1413pm2.21d 118 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
15143expia 1264 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
16 ax-1 6 . . . . . 6 (𝑥 = 𝑦 → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
1716a1i 11 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 = 𝑦 → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
18 r1sdom 8584 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑅1𝑦) ≺ (𝑅1𝑥))
19 breq2 4619 . . . . . . . . . 10 ((𝑅1𝑥) = (𝑅1𝑦) → ((𝑅1𝑦) ≺ (𝑅1𝑥) ↔ (𝑅1𝑦) ≺ (𝑅1𝑦)))
2018, 19syl5ibcom 235 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦𝑥) → ((𝑅1𝑥) = (𝑅1𝑦) → (𝑅1𝑦) ≺ (𝑅1𝑦)))
218, 20mtoi 190 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
22213adant2 1078 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦𝑥) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
2322pm2.21d 118 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦𝑥) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
24233expia 1264 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦𝑥 → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
2515, 17, 243jaod 1389 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
267, 25mpd 15 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
2726rgen2a 2971 . 2 𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)
28 dff13 6469 . 2 (𝑅1:On–1-1→V ↔ (𝑅1:On⟶V ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
293, 27, 28mpbir2an 954 1 𝑅1:On–1-1→V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3o 1035  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186   class class class wbr 4615  Ord word 5683  Oncon0 5684   Fn wfn 5844  wf 5845  1-1wf1 5846  cfv 5849  csdm 7901  𝑅1cr1 8572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-r1 8574
This theorem is referenced by:  tskinf  9538  grothomex  9598  rankeq1o  31941  elhf  31944  hfninf  31956
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