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Theorem r111 8803
 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 8795 . . 3 𝑅1 Fn On
2 dffn2 6200 . . 3 (𝑅1 Fn On ↔ 𝑅1:On⟶V)
31, 2mpbi 220 . 2 𝑅1:On⟶V
4 eloni 5886 . . . . 5 (𝑥 ∈ On → Ord 𝑥)
5 eloni 5886 . . . . 5 (𝑦 ∈ On → Ord 𝑦)
6 ordtri3or 5908 . . . . 5 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
74, 5, 6syl2an 495 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
8 sdomirr 8254 . . . . . . . . 9 ¬ (𝑅1𝑦) ≺ (𝑅1𝑦)
9 r1sdom 8802 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝑥𝑦) → (𝑅1𝑥) ≺ (𝑅1𝑦))
10 breq1 4799 . . . . . . . . . 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 1124 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
1413pm2.21d 118 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
15143expia 1114 . . . . 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 8802 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑅1𝑦) ≺ (𝑅1𝑥))
19 breq2 4800 . . . . . . . . . 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 1125 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦𝑥) → ¬ (𝑅1𝑥) = (𝑅1𝑦))
2322pm2.21d 118 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦𝑥) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
24233expia 1114 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦𝑥 → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
2515, 17, 243jaod 1533 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
267, 25mpd 15 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦))
2726rgen2a 3107 . 2 𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)
28 dff13 6667 . 2 (𝑅1:On–1-1→V ↔ (𝑅1:On⟶V ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1𝑥) = (𝑅1𝑦) → 𝑥 = 𝑦)))
293, 27, 28mpbir2an 993 1 𝑅1:On–1-1→V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∨ w3o 1071   ∧ w3a 1072   = wceq 1624   ∈ wcel 2131  ∀wral 3042  Vcvv 3332   class class class wbr 4796  Ord word 5875  Oncon0 5876   Fn wfn 6036  ⟶wf 6037  –1-1→wf1 6038  ‘cfv 6041   ≺ csdm 8112  𝑅1cr1 8790 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-r1 8792 This theorem is referenced by:  tskinf  9775  grothomex  9835  rankeq1o  32576  elhf  32579  hfninf  32591
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