MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  phplem2 Structured version   Visualization version   GIF version

Theorem phplem2 9208
Description: Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) Avoid ax-pow 5364. (Revised by BTernaryTau, 4-Nov-2024.)
Hypothesis
Ref Expression
phplem2.1 𝐴 ∈ V
Assertion
Ref Expression
phplem2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵𝐴𝐵))

Proof of Theorem phplem2
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 bren 8949 . 2 (suc 𝐴 ≈ suc 𝐵 ↔ ∃𝑓 𝑓:suc 𝐴1-1-onto→suc 𝐵)
2 f1of1 6833 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵𝑓:suc 𝐴1-1→suc 𝐵)
3 nnfi 9167 . . . . . . . . 9 (𝐴 ∈ ω → 𝐴 ∈ Fin)
4 sssucid 6445 . . . . . . . . . 10 𝐴 ⊆ suc 𝐴
5 f1imaenfi 9198 . . . . . . . . . 10 ((𝑓:suc 𝐴1-1→suc 𝐵𝐴 ⊆ suc 𝐴𝐴 ∈ Fin) → (𝑓𝐴) ≈ 𝐴)
64, 5mp3an2 1450 . . . . . . . . 9 ((𝑓:suc 𝐴1-1→suc 𝐵𝐴 ∈ Fin) → (𝑓𝐴) ≈ 𝐴)
72, 3, 6syl2anr 598 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (𝑓𝐴) ≈ 𝐴)
8 ensymfib 9187 . . . . . . . . . 10 (𝐴 ∈ Fin → (𝐴 ≈ (𝑓𝐴) ↔ (𝑓𝐴) ≈ 𝐴))
93, 8syl 17 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 ≈ (𝑓𝐴) ↔ (𝑓𝐴) ≈ 𝐴))
109adantr 482 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (𝐴 ≈ (𝑓𝐴) ↔ (𝑓𝐴) ≈ 𝐴))
117, 10mpbird 257 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → 𝐴 ≈ (𝑓𝐴))
12 nnord 7863 . . . . . . . . . 10 (𝐴 ∈ ω → Ord 𝐴)
13 orddif 6461 . . . . . . . . . 10 (Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))
1412, 13syl 17 . . . . . . . . 9 (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴}))
1514imaeq2d 6060 . . . . . . . 8 (𝐴 ∈ ω → (𝑓𝐴) = (𝑓 “ (suc 𝐴 ∖ {𝐴})))
16 f1ofn 6835 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵𝑓 Fn suc 𝐴)
17 phplem2.1 . . . . . . . . . . . 12 𝐴 ∈ V
1817sucid 6447 . . . . . . . . . . 11 𝐴 ∈ suc 𝐴
19 fnsnfv 6971 . . . . . . . . . . 11 ((𝑓 Fn suc 𝐴𝐴 ∈ suc 𝐴) → {(𝑓𝐴)} = (𝑓 “ {𝐴}))
2016, 18, 19sylancl 587 . . . . . . . . . 10 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → {(𝑓𝐴)} = (𝑓 “ {𝐴}))
2120difeq2d 4123 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → ((𝑓 “ suc 𝐴) ∖ {(𝑓𝐴)}) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
22 imadmrn 6070 . . . . . . . . . . . 12 (𝑓 “ dom 𝑓) = ran 𝑓
2322eqcomi 2742 . . . . . . . . . . 11 ran 𝑓 = (𝑓 “ dom 𝑓)
24 f1ofo 6841 . . . . . . . . . . . 12 (𝑓:suc 𝐴1-1-onto→suc 𝐵𝑓:suc 𝐴onto→suc 𝐵)
25 forn 6809 . . . . . . . . . . . 12 (𝑓:suc 𝐴onto→suc 𝐵 → ran 𝑓 = suc 𝐵)
2624, 25syl 17 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → ran 𝑓 = suc 𝐵)
27 f1odm 6838 . . . . . . . . . . . 12 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → dom 𝑓 = suc 𝐴)
2827imaeq2d 6060 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓 “ dom 𝑓) = (𝑓 “ suc 𝐴))
2923, 26, 283eqtr3a 2797 . . . . . . . . . 10 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → suc 𝐵 = (𝑓 “ suc 𝐴))
3029difeq1d 4122 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (suc 𝐵 ∖ {(𝑓𝐴)}) = ((𝑓 “ suc 𝐴) ∖ {(𝑓𝐴)}))
31 dff1o3 6840 . . . . . . . . . 10 (𝑓:suc 𝐴1-1-onto→suc 𝐵 ↔ (𝑓:suc 𝐴onto→suc 𝐵 ∧ Fun 𝑓))
32 imadif 6633 . . . . . . . . . 10 (Fun 𝑓 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
3331, 32simplbiim 506 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
3421, 30, 333eqtr4rd 2784 . . . . . . . 8 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = (suc 𝐵 ∖ {(𝑓𝐴)}))
3515, 34sylan9eq 2793 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (𝑓𝐴) = (suc 𝐵 ∖ {(𝑓𝐴)}))
3611, 35breqtrd 5175 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}))
37 fnfvelrn 7083 . . . . . . . . . . . 12 ((𝑓 Fn suc 𝐴𝐴 ∈ suc 𝐴) → (𝑓𝐴) ∈ ran 𝑓)
3816, 18, 37sylancl 587 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓𝐴) ∈ ran 𝑓)
3925eleq2d 2820 . . . . . . . . . . . 12 (𝑓:suc 𝐴onto→suc 𝐵 → ((𝑓𝐴) ∈ ran 𝑓 ↔ (𝑓𝐴) ∈ suc 𝐵))
4024, 39syl 17 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → ((𝑓𝐴) ∈ ran 𝑓 ↔ (𝑓𝐴) ∈ suc 𝐵))
4138, 40mpbid 231 . . . . . . . . . 10 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓𝐴) ∈ suc 𝐵)
42 phplem1 9207 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ (𝑓𝐴) ∈ suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}))
4341, 42sylan2 594 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}))
44 nnfi 9167 . . . . . . . . . . 11 (𝐵 ∈ ω → 𝐵 ∈ Fin)
45 ensymfib 9187 . . . . . . . . . . 11 (𝐵 ∈ Fin → (𝐵 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ↔ (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵))
4644, 45syl 17 . . . . . . . . . 10 (𝐵 ∈ ω → (𝐵 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ↔ (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵))
4746adantr 482 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (𝐵 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ↔ (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵))
4843, 47mpbid 231 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵)
49 entrfil 9188 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵) → 𝐴𝐵)
503, 49syl3an1 1164 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵) → 𝐴𝐵)
5148, 50syl3an3 1166 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵)) → 𝐴𝐵)
52513expa 1119 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)})) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵)) → 𝐴𝐵)
5336, 52syldanl 603 . . . . 5 (((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵)) → 𝐴𝐵)
5453anandirs 678 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → 𝐴𝐵)
5554ex 414 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑓:suc 𝐴1-1-onto→suc 𝐵𝐴𝐵))
5655exlimdv 1937 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑓 𝑓:suc 𝐴1-1-onto→suc 𝐵𝐴𝐵))
571, 56biimtrid 241 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  Vcvv 3475  cdif 3946  wss 3949  {csn 4629   class class class wbr 5149  ccnv 5676  dom cdm 5677  ran crn 5678  cima 5680  Ord word 6364  suc csuc 6367  Fun wfun 6538   Fn wfn 6539  1-1wf1 6541  ontowfo 6542  1-1-ontowf1o 6543  cfv 6544  ωcom 7855  cen 8936  Fincfn 8939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-en 8940  df-fin 8943
This theorem is referenced by:  nneneq  9209
  Copyright terms: Public domain W3C validator