Theorem fin1a2lem2 9812

Description: Lemma for fin1a2 9826. (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.

Step	Hyp	Ref	Expression
1		fin1a2lem.a	. . 3 ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
2		suceloni 7508	. . 3 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
3	1, 2	fmpti 6853	. 2 ⊢ 𝑆:On⟶On
4	1	fin1a2lem1 9811	. . . . . 6 ⊢ (𝑎 ∈ On → (𝑆‘𝑎) = suc 𝑎)
5	1	fin1a2lem1 9811	. . . . . 6 ⊢ (𝑏 ∈ On → (𝑆‘𝑏) = suc 𝑏)
6	4, 5	eqeqan12d 2815	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ suc 𝑎 = suc 𝑏))
7		suc11 6262	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (suc 𝑎 = suc 𝑏 ↔ 𝑎 = 𝑏))
8	6, 7	bitrd 282	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ 𝑎 = 𝑏))
9	8	biimpd 232	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏))
10	9	rgen2 3168	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏)
11		dff13 6991	. 2 ⊢ (𝑆:On–1-1→On ↔ (𝑆:On⟶On ∧ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏)))
12	3, 10, 11	mpbir2an 710	1 ⊢ 𝑆:On–1-1→On

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ↦ cmpt 5110  Oncon0 6159  suc csuc 6161  ⟶wf 6320  –1-1→wf1 6321  ‘cfv 6324

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fv 6332

This theorem is referenced by:  fin1a2lem6  9816