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Theorem phplem2 9245
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 5365. (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 8995 . 2 (suc 𝐴 ≈ suc 𝐵 ↔ ∃𝑓 𝑓:suc 𝐴1-1-onto→suc 𝐵)
2 f1of1 6847 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵𝑓:suc 𝐴1-1→suc 𝐵)
3 nnfi 9207 . . . . . . . . 9 (𝐴 ∈ ω → 𝐴 ∈ Fin)
4 sssucid 6464 . . . . . . . . . 10 𝐴 ⊆ suc 𝐴
5 f1imaenfi 9235 . . . . . . . . . 10 ((𝑓:suc 𝐴1-1→suc 𝐵𝐴 ⊆ suc 𝐴𝐴 ∈ Fin) → (𝑓𝐴) ≈ 𝐴)
64, 5mp3an2 1451 . . . . . . . . 9 ((𝑓:suc 𝐴1-1→suc 𝐵𝐴 ∈ Fin) → (𝑓𝐴) ≈ 𝐴)
72, 3, 6syl2anr 597 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (𝑓𝐴) ≈ 𝐴)
8 ensymfib 9224 . . . . . . . . . 10 (𝐴 ∈ Fin → (𝐴 ≈ (𝑓𝐴) ↔ (𝑓𝐴) ≈ 𝐴))
93, 8syl 17 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 ≈ (𝑓𝐴) ↔ (𝑓𝐴) ≈ 𝐴))
109adantr 480 . . . . . . . 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 7895 . . . . . . . . . 10 (𝐴 ∈ ω → Ord 𝐴)
13 orddif 6480 . . . . . . . . . 10 (Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))
1412, 13syl 17 . . . . . . . . 9 (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴}))
1514imaeq2d 6078 . . . . . . . 8 (𝐴 ∈ ω → (𝑓𝐴) = (𝑓 “ (suc 𝐴 ∖ {𝐴})))
16 f1ofn 6849 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵𝑓 Fn suc 𝐴)
17 phplem2.1 . . . . . . . . . . . 12 𝐴 ∈ V
1817sucid 6466 . . . . . . . . . . 11 𝐴 ∈ suc 𝐴
19 fnsnfv 6988 . . . . . . . . . . 11 ((𝑓 Fn suc 𝐴𝐴 ∈ suc 𝐴) → {(𝑓𝐴)} = (𝑓 “ {𝐴}))
2016, 18, 19sylancl 586 . . . . . . . . . 10 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → {(𝑓𝐴)} = (𝑓 “ {𝐴}))
2120difeq2d 4126 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → ((𝑓 “ suc 𝐴) ∖ {(𝑓𝐴)}) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
22 imadmrn 6088 . . . . . . . . . . . 12 (𝑓 “ dom 𝑓) = ran 𝑓
2322eqcomi 2746 . . . . . . . . . . 11 ran 𝑓 = (𝑓 “ dom 𝑓)
24 f1ofo 6855 . . . . . . . . . . . 12 (𝑓:suc 𝐴1-1-onto→suc 𝐵𝑓:suc 𝐴onto→suc 𝐵)
25 forn 6823 . . . . . . . . . . . 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 6852 . . . . . . . . . . . 12 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → dom 𝑓 = suc 𝐴)
2827imaeq2d 6078 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓 “ dom 𝑓) = (𝑓 “ suc 𝐴))
2923, 26, 283eqtr3a 2801 . . . . . . . . . 10 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → suc 𝐵 = (𝑓 “ suc 𝐴))
3029difeq1d 4125 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (suc 𝐵 ∖ {(𝑓𝐴)}) = ((𝑓 “ suc 𝐴) ∖ {(𝑓𝐴)}))
31 dff1o3 6854 . . . . . . . . . 10 (𝑓:suc 𝐴1-1-onto→suc 𝐵 ↔ (𝑓:suc 𝐴onto→suc 𝐵 ∧ Fun 𝑓))
32 imadif 6650 . . . . . . . . . 10 (Fun 𝑓 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
3331, 32simplbiim 504 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
3421, 30, 333eqtr4rd 2788 . . . . . . . 8 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = (suc 𝐵 ∖ {(𝑓𝐴)}))
3515, 34sylan9eq 2797 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (𝑓𝐴) = (suc 𝐵 ∖ {(𝑓𝐴)}))
3611, 35breqtrd 5169 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}))
37 fnfvelrn 7100 . . . . . . . . . . . 12 ((𝑓 Fn suc 𝐴𝐴 ∈ suc 𝐴) → (𝑓𝐴) ∈ ran 𝑓)
3816, 18, 37sylancl 586 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓𝐴) ∈ ran 𝑓)
3925eleq2d 2827 . . . . . . . . . . . 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 232 . . . . . . . . . 10 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓𝐴) ∈ suc 𝐵)
42 phplem1 9244 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ (𝑓𝐴) ∈ suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}))
4341, 42sylan2 593 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}))
44 nnfi 9207 . . . . . . . . . . 11 (𝐵 ∈ ω → 𝐵 ∈ Fin)
45 ensymfib 9224 . . . . . . . . . . 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 480 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (𝐵 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ↔ (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵))
4843, 47mpbid 232 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵)
49 entrfil 9225 . . . . . . . . 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 602 . . . . 5 (((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵)) → 𝐴𝐵)
5453anandirs 679 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → 𝐴𝐵)
5554ex 412 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑓:suc 𝐴1-1-onto→suc 𝐵𝐴𝐵))
5655exlimdv 1933 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑓 𝑓:suc 𝐴1-1-onto→suc 𝐵𝐴𝐵))
571, 56biimtrid 242 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cdif 3948  wss 3951  {csn 4626   class class class wbr 5143  ccnv 5684  dom cdm 5685  ran crn 5686  cima 5688  Ord word 6383  suc csuc 6386  Fun wfun 6555   Fn wfn 6556  1-1wf1 6558  ontowfo 6559  1-1-ontowf1o 6560  cfv 6561  ωcom 7887  cen 8982  Fincfn 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-en 8986  df-fin 8989
This theorem is referenced by:  nneneq  9246
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