Theorem r111 9674

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.

Step	Hyp	Ref	Expression
1		r1fnon 9666	. . 3 ⊢ 𝑅1 Fn On
2		dffn2 6659	. . 3 ⊢ (𝑅1 Fn On ↔ 𝑅1:On⟶V)
3	1, 2	mpbi 230	. 2 ⊢ 𝑅1:On⟶V
4		eloni 6322	. . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥)
5		eloni 6322	. . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦)
6		ordtri3or 6344	. . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
7	4, 5, 6	syl2an 596	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
8		sdomirr 9033	. . . . . . . . 9 ⊢ ¬ (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)
9		r1sdom 9673	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → (𝑅1‘𝑥) ≺ (𝑅1‘𝑦))
10		breq1 5096	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑥) = (𝑅1‘𝑦) → ((𝑅1‘𝑥) ≺ (𝑅1‘𝑦) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
11	9, 10	syl5ibcom 245	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
12	8, 11	mtoi 199	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → ¬ (𝑅1‘𝑥) = (𝑅1‘𝑦))
13	12	3adant1 1130	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → ¬ (𝑅1‘𝑥) = (𝑅1‘𝑦))
14	13	pm2.21d 121	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦))
15	14	3expia 1121	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)))
16		ax-1 6	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦))
17	16	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 = 𝑦 → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)))
18		r1sdom 9673	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑥))
19		breq2 5097	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑥) = (𝑅1‘𝑦) → ((𝑅1‘𝑦) ≺ (𝑅1‘𝑥) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
20	18, 19	syl5ibcom 245	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
21	8, 20	mtoi 199	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝑅1‘𝑥) = (𝑅1‘𝑦))
22	21	3adant2 1131	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝑅1‘𝑥) = (𝑅1‘𝑦))
23	22	pm2.21d 121	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦))
24	23	3expia 1121	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ 𝑥 → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)))
25	15, 17, 24	3jaod 1431	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)))
26	7, 25	mpd 15	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦))
27	26	rgen2 3172	. 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)
28		dff13 7194	. 2 ⊢ (𝑅1:On–1-1→V ↔ (𝑅1:On⟶V ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)))
29	3, 27, 28	mpbir2an 711	1 ⊢ 𝑅1:On–1-1→V

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ w3o 1085   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   class class class wbr 5093  Ord word 6311  Oncon0 6312   Fn wfn 6482  ⟶wf 6483  –1-1→wf1 6484  ‘cfv 6487   ≺ csdm 8874  𝑅₁cr1 9661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-r1 9663

This theorem is referenced by:  tskinf  10666  grothomex  10726  rankeq1o  36222  elhf  36225  hfninf  36237