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Theorem fin1a2lem2 9800
Description: Lemma for fin1a2 9814. (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 7503 . . 3 (𝑥 ∈ On → suc 𝑥 ∈ On)
31, 2fmpti 6849 . 2 𝑆:On⟶On
41fin1a2lem1 9799 . . . . . 6 (𝑎 ∈ On → (𝑆𝑎) = suc 𝑎)
51fin1a2lem1 9799 . . . . . 6 (𝑏 ∈ On → (𝑆𝑏) = suc 𝑏)
64, 5eqeqan12d 2838 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆𝑎) = (𝑆𝑏) ↔ suc 𝑎 = suc 𝑏))
7 suc11 6267 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (suc 𝑎 = suc 𝑏𝑎 = 𝑏))
86, 7bitrd 282 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆𝑎) = (𝑆𝑏) ↔ 𝑎 = 𝑏))
98biimpd 232 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆𝑎) = (𝑆𝑏) → 𝑎 = 𝑏))
109rgen2 3191 . 2 𝑎 ∈ On ∀𝑏 ∈ On ((𝑆𝑎) = (𝑆𝑏) → 𝑎 = 𝑏)
11 dff13 6987 . 2 (𝑆:On–1-1→On ↔ (𝑆:On⟶On ∧ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆𝑎) = (𝑆𝑏) → 𝑎 = 𝑏)))
123, 10, 11mpbir2an 710 1 𝑆:On–1-1→On
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3126  cmpt 5119  Oncon0 6164  suc csuc 6166  wf 6324  1-1wf1 6325  cfv 6328
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ord 6167  df-on 6168  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fv 6336
This theorem is referenced by:  fin1a2lem6  9804
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