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Theorem fin1a2lem2 9106
 Description: Lemma for fin1a2 9120. (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 6905 . . 3 (𝑥 ∈ On → suc 𝑥 ∈ On)
31, 2fmpti 6291 . 2 𝑆:On⟶On
41fin1a2lem1 9105 . . . . . 6 (𝑎 ∈ On → (𝑆𝑎) = suc 𝑎)
51fin1a2lem1 9105 . . . . . 6 (𝑏 ∈ On → (𝑆𝑏) = suc 𝑏)
64, 5eqeqan12d 2626 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆𝑎) = (𝑆𝑏) ↔ suc 𝑎 = suc 𝑏))
7 suc11 5748 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (suc 𝑎 = suc 𝑏𝑎 = 𝑏))
86, 7bitrd 267 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆𝑎) = (𝑆𝑏) ↔ 𝑎 = 𝑏))
98biimpd 218 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆𝑎) = (𝑆𝑏) → 𝑎 = 𝑏))
109rgen2a 2960 . 2 𝑎 ∈ On ∀𝑏 ∈ On ((𝑆𝑎) = (𝑆𝑏) → 𝑎 = 𝑏)
11 dff13 6416 . 2 (𝑆:On–1-1→On ↔ (𝑆:On⟶On ∧ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆𝑎) = (𝑆𝑏) → 𝑎 = 𝑏)))
123, 10, 11mpbir2an 957 1 𝑆:On–1-1→On
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ↦ cmpt 4643  Oncon0 5640  suc csuc 5642  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804 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-sep 4709  ax-nul 4717  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-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  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-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812 This theorem is referenced by:  fin1a2lem6  9110
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