Theorem r111 9710

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 9702	. . 3 ⊢ 𝑅1 Fn On
2		dffn2 6670	. . 3 ⊢ (𝑅1 Fn On ↔ 𝑅1:On⟶V)
3	1, 2	mpbi 229	. 2 ⊢ 𝑅1:On⟶V
4		eloni 6327	. . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥)
5		eloni 6327	. . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦)
6		ordtri3or 6349	. . . . 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 9057	. . . . . . . . 9 ⊢ ¬ (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)
9		r1sdom 9709	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → (𝑅1‘𝑥) ≺ (𝑅1‘𝑦))
10		breq1 5108	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑥) = (𝑅1‘𝑦) → ((𝑅1‘𝑥) ≺ (𝑅1‘𝑦) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
11	9, 10	syl5ibcom 244	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
12	8, 11	mtoi 198	. . . . . . . 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 9709	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑥))
19		breq2 5109	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑥) = (𝑅1‘𝑦) → ((𝑅1‘𝑦) ≺ (𝑅1‘𝑥) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
20	18, 19	syl5ibcom 244	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑅1‘𝑥) = (𝑅1‘𝑦) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑦)))
21	8, 20	mtoi 198	. . . . . . . 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 1428	. . . 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 3194	. 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)
28		dff13 7201	. 2 ⊢ (𝑅1:On–1-1→V ↔ (𝑅1:On⟶V ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑅1‘𝑥) = (𝑅1‘𝑦) → 𝑥 = 𝑦)))
29	3, 27, 28	mpbir2an 709	1 ⊢ 𝑅1:On–1-1→V

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ w3o 1086   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  Vcvv 3445   class class class wbr 5105  Ord word 6316  Oncon0 6317   Fn wfn 6491  ⟶wf 6492  –1-1→wf1 6493  ‘cfv 6496   ≺ csdm 8881  𝑅₁cr1 9697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7359  df-2nd 7921  df-frecs 8211  df-wrecs 8242  df-recs 8316  df-rdg 8355  df-er 8647  df-en 8883  df-dom 8884  df-sdom 8885  df-r1 9699

This theorem is referenced by:  tskinf  10704  grothomex  10764  rankeq1o  34747  elhf  34750  hfninf  34762