MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin1a2lem2 Structured version   Visualization version   GIF version

Theorem fin1a2lem2 9509
Description: Lemma for fin1a2 9523. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypothesis
Ref Expression
fin1a2lem.a 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
Assertion
Ref Expression
fin1a2lem2 𝑆:On–1-1→On

Proof of Theorem fin1a2lem2
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fin1a2lem.a . . 3 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
2 suceloni 7245 . . 3 (𝑥 ∈ On → suc 𝑥 ∈ On)
31, 2fmpti 6606 . 2 𝑆:On⟶On
41fin1a2lem1 9508 . . . . . 6 (𝑎 ∈ On → (𝑆𝑎) = suc 𝑎)
51fin1a2lem1 9508 . . . . . 6 (𝑏 ∈ On → (𝑆𝑏) = suc 𝑏)
64, 5eqeqan12d 2813 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆𝑎) = (𝑆𝑏) ↔ suc 𝑎 = suc 𝑏))
7 suc11 6042 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (suc 𝑎 = suc 𝑏𝑎 = 𝑏))
86, 7bitrd 271 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆𝑎) = (𝑆𝑏) ↔ 𝑎 = 𝑏))
98biimpd 221 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆𝑎) = (𝑆𝑏) → 𝑎 = 𝑏))
109rgen2a 3156 . 2 𝑎 ∈ On ∀𝑏 ∈ On ((𝑆𝑎) = (𝑆𝑏) → 𝑎 = 𝑏)
11 dff13 6738 . 2 (𝑆:On–1-1→On ↔ (𝑆:On⟶On ∧ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆𝑎) = (𝑆𝑏) → 𝑎 = 𝑏)))
123, 10, 11mpbir2an 703 1 𝑆:On–1-1→On
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3087  cmpt 4920  Oncon0 5939  suc csuc 5941  wf 6095  1-1wf1 6096  cfv 6099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-ord 5942  df-on 5943  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fv 6107
This theorem is referenced by:  fin1a2lem6  9513
  Copyright terms: Public domain W3C validator