Theorem r111 8521

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 8513	. . 3 ⊢ 𝑅1 Fn On
2		dffn2 5960	. . 3 ⊢ (𝑅1 Fn On ↔ 𝑅1:On⟶V)
3	1, 2	mpbi 219	. 2 ⊢ 𝑅1:On⟶V
4		eloni 5650	. . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥)
5		eloni 5650	. . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦)
6		ordtri3or 5672	. . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
7	4, 5, 6	syl2an 493	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
8		sdomirr 7982	. . . . . . . . 9 ⊢ ¬ (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)
9		r1sdom 8520	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → (𝑅1‘𝑥) ≺ (𝑅1‘𝑦))
10		breq1 4586	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑥) = (𝑅1‘𝑦) → ((𝑅1‘𝑥) ≺ (𝑅1‘𝑦) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
11	9, 10	syl5ibcom 234	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
12	8, 11	mtoi 189	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → ¬ (𝑅1‘𝑥) = (𝑅1‘𝑦))
13	12	3adant1 1072	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → ¬ (𝑅1‘𝑥) = (𝑅1‘𝑦))
14	13	pm2.21d 117	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦))
15	14	3expia 1259	. . . . 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 8520	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑥))
19		breq2 4587	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑥) = (𝑅1‘𝑦) → ((𝑅1‘𝑦) ≺ (𝑅1‘𝑥) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
20	18, 19	syl5ibcom 234	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
21	8, 20	mtoi 189	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝑅1‘𝑥) = (𝑅1‘𝑦))
22	21	3adant2 1073	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝑅1‘𝑥) = (𝑅1‘𝑦))
23	22	pm2.21d 117	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦))
24	23	3expia 1259	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ 𝑥 → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)))
25	15, 17, 24	3jaod 1384	. . . 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	rgen2a 2960	. 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)
28		dff13 6416	. 2 ⊢ (𝑅1:On–1-1→V ↔ (𝑅1:On⟶V ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)))
29	3, 27, 28	mpbir2an 957	1 ⊢ 𝑅1:On–1-1→V

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   class class class wbr 4583  Ord word 5639  Oncon0 5640   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804   ≺ csdm 7840  𝑅₁cr1 8508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-r1 8510

This theorem is referenced by:  tskinf  9470  grothomex  9530  rankeq1o  31448  elhf  31451  hfninf  31463