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Theorem r111 9770
Description: The cumulative hierarchy is a one-to-one function. (Contributed by Mario Carneiro, 19-Apr-2013.)
Assertion
Ref Expression
r111 𝑅1:On–1-1β†’V

Proof of Theorem r111
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1fnon 9762 . . 3 𝑅1 Fn On
2 dffn2 6720 . . 3 (𝑅1 Fn On ↔ 𝑅1:On⟢V)
31, 2mpbi 229 . 2 𝑅1:On⟢V
4 eloni 6375 . . . . 5 (π‘₯ ∈ On β†’ Ord π‘₯)
5 eloni 6375 . . . . 5 (𝑦 ∈ On β†’ Ord 𝑦)
6 ordtri3or 6397 . . . . 5 ((Ord π‘₯ ∧ Ord 𝑦) β†’ (π‘₯ ∈ 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 ∈ π‘₯))
74, 5, 6syl2an 597 . . . 4 ((π‘₯ ∈ On ∧ 𝑦 ∈ On) β†’ (π‘₯ ∈ 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 ∈ π‘₯))
8 sdomirr 9114 . . . . . . . . 9 Β¬ (𝑅1β€˜π‘¦) β‰Ί (𝑅1β€˜π‘¦)
9 r1sdom 9769 . . . . . . . . . 10 ((𝑦 ∈ On ∧ π‘₯ ∈ 𝑦) β†’ (𝑅1β€˜π‘₯) β‰Ί (𝑅1β€˜π‘¦))
10 breq1 5152 . . . . . . . . . 10 ((𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦) β†’ ((𝑅1β€˜π‘₯) β‰Ί (𝑅1β€˜π‘¦) ↔ (𝑅1β€˜π‘¦) β‰Ί (𝑅1β€˜π‘¦)))
119, 10syl5ibcom 244 . . . . . . . . 9 ((𝑦 ∈ On ∧ π‘₯ ∈ 𝑦) β†’ ((𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦) β†’ (𝑅1β€˜π‘¦) β‰Ί (𝑅1β€˜π‘¦)))
128, 11mtoi 198 . . . . . . . 8 ((𝑦 ∈ On ∧ π‘₯ ∈ 𝑦) β†’ Β¬ (𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦))
13123adant1 1131 . . . . . . 7 ((π‘₯ ∈ On ∧ 𝑦 ∈ On ∧ π‘₯ ∈ 𝑦) β†’ Β¬ (𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦))
1413pm2.21d 121 . . . . . 6 ((π‘₯ ∈ On ∧ 𝑦 ∈ On ∧ π‘₯ ∈ 𝑦) β†’ ((𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦) β†’ π‘₯ = 𝑦))
15143expia 1122 . . . . 5 ((π‘₯ ∈ On ∧ 𝑦 ∈ On) β†’ (π‘₯ ∈ 𝑦 β†’ ((𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦) β†’ π‘₯ = 𝑦)))
16 ax-1 6 . . . . . 6 (π‘₯ = 𝑦 β†’ ((𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦) β†’ π‘₯ = 𝑦))
1716a1i 11 . . . . 5 ((π‘₯ ∈ On ∧ 𝑦 ∈ On) β†’ (π‘₯ = 𝑦 β†’ ((𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦) β†’ π‘₯ = 𝑦)))
18 r1sdom 9769 . . . . . . . . . 10 ((π‘₯ ∈ On ∧ 𝑦 ∈ π‘₯) β†’ (𝑅1β€˜π‘¦) β‰Ί (𝑅1β€˜π‘₯))
19 breq2 5153 . . . . . . . . . 10 ((𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦) β†’ ((𝑅1β€˜π‘¦) β‰Ί (𝑅1β€˜π‘₯) ↔ (𝑅1β€˜π‘¦) β‰Ί (𝑅1β€˜π‘¦)))
2018, 19syl5ibcom 244 . . . . . . . . 9 ((π‘₯ ∈ On ∧ 𝑦 ∈ π‘₯) β†’ ((𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦) β†’ (𝑅1β€˜π‘¦) β‰Ί (𝑅1β€˜π‘¦)))
218, 20mtoi 198 . . . . . . . 8 ((π‘₯ ∈ On ∧ 𝑦 ∈ π‘₯) β†’ Β¬ (𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦))
22213adant2 1132 . . . . . . 7 ((π‘₯ ∈ On ∧ 𝑦 ∈ On ∧ 𝑦 ∈ π‘₯) β†’ Β¬ (𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦))
2322pm2.21d 121 . . . . . 6 ((π‘₯ ∈ On ∧ 𝑦 ∈ On ∧ 𝑦 ∈ π‘₯) β†’ ((𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦) β†’ π‘₯ = 𝑦))
24233expia 1122 . . . . 5 ((π‘₯ ∈ On ∧ 𝑦 ∈ On) β†’ (𝑦 ∈ π‘₯ β†’ ((𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦) β†’ π‘₯ = 𝑦)))
2515, 17, 243jaod 1429 . . . 4 ((π‘₯ ∈ On ∧ 𝑦 ∈ On) β†’ ((π‘₯ ∈ 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 ∈ π‘₯) β†’ ((𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦) β†’ π‘₯ = 𝑦)))
267, 25mpd 15 . . 3 ((π‘₯ ∈ On ∧ 𝑦 ∈ On) β†’ ((𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦) β†’ π‘₯ = 𝑦))
2726rgen2 3198 . 2 βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On ((𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦) β†’ π‘₯ = 𝑦)
28 dff13 7254 . 2 (𝑅1:On–1-1β†’V ↔ (𝑅1:On⟢V ∧ βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On ((𝑅1β€˜π‘₯) = (𝑅1β€˜π‘¦) β†’ π‘₯ = 𝑦)))
293, 27, 28mpbir2an 710 1 𝑅1:On–1-1β†’V
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∨ w3o 1087   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  Vcvv 3475   class class class wbr 5149  Ord word 6364  Oncon0 6365   Fn wfn 6539  βŸΆwf 6540  β€“1-1β†’wf1 6541  β€˜cfv 6544   β‰Ί csdm 8938  π‘…1cr1 9757
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-r1 9759
This theorem is referenced by:  tskinf  10764  grothomex  10824  rankeq1o  35143  elhf  35146  hfninf  35158
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