Theorem r111 9728

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 9720	. . 3 ⊢ 𝑅1 Fn On
2		dffn2 6690	. . 3 ⊢ (𝑅1 Fn On ↔ 𝑅1:On⟶V)
3	1, 2	mpbi 230	. 2 ⊢ 𝑅1:On⟶V
4		eloni 6342	. . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥)
5		eloni 6342	. . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦)
6		ordtri3or 6364	. . . . 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 9078	. . . . . . . . 9 ⊢ ¬ (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)
9		r1sdom 9727	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → (𝑅1‘𝑥) ≺ (𝑅1‘𝑦))
10		breq1 5110	. . . . . . . . . 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 9727	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑥))
19		breq2 5111	. . . . . . . . . 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 3177	. 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)
28		dff13 7229	. 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 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   class class class wbr 5107  Ord word 6331  Oncon0 6332   Fn wfn 6506  ⟶wf 6507  –1-1→wf1 6508  ‘cfv 6511   ≺ csdm 8917  𝑅₁cr1 9715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-r1 9717

This theorem is referenced by:  tskinf  10722  grothomex  10782  rankeq1o  36159  elhf  36162  hfninf  36174