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Theorem phplem2 9242
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 5369. (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 8984 . 2 (suc 𝐴 ≈ suc 𝐵 ↔ ∃𝑓 𝑓:suc 𝐴1-1-onto→suc 𝐵)
2 f1of1 6842 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵𝑓:suc 𝐴1-1→suc 𝐵)
3 nnfi 9205 . . . . . . . . 9 (𝐴 ∈ ω → 𝐴 ∈ Fin)
4 sssucid 6456 . . . . . . . . . 10 𝐴 ⊆ suc 𝐴
5 f1imaenfi 9232 . . . . . . . . . 10 ((𝑓:suc 𝐴1-1→suc 𝐵𝐴 ⊆ suc 𝐴𝐴 ∈ Fin) → (𝑓𝐴) ≈ 𝐴)
64, 5mp3an2 1446 . . . . . . . . 9 ((𝑓:suc 𝐴1-1→suc 𝐵𝐴 ∈ Fin) → (𝑓𝐴) ≈ 𝐴)
72, 3, 6syl2anr 595 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (𝑓𝐴) ≈ 𝐴)
8 ensymfib 9221 . . . . . . . . . 10 (𝐴 ∈ Fin → (𝐴 ≈ (𝑓𝐴) ↔ (𝑓𝐴) ≈ 𝐴))
93, 8syl 17 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 ≈ (𝑓𝐴) ↔ (𝑓𝐴) ≈ 𝐴))
109adantr 479 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (𝐴 ≈ (𝑓𝐴) ↔ (𝑓𝐴) ≈ 𝐴))
117, 10mpbird 256 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → 𝐴 ≈ (𝑓𝐴))
12 nnord 7884 . . . . . . . . . 10 (𝐴 ∈ ω → Ord 𝐴)
13 orddif 6472 . . . . . . . . . 10 (Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))
1412, 13syl 17 . . . . . . . . 9 (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴}))
1514imaeq2d 6069 . . . . . . . 8 (𝐴 ∈ ω → (𝑓𝐴) = (𝑓 “ (suc 𝐴 ∖ {𝐴})))
16 f1ofn 6844 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵𝑓 Fn suc 𝐴)
17 phplem2.1 . . . . . . . . . . . 12 𝐴 ∈ V
1817sucid 6458 . . . . . . . . . . 11 𝐴 ∈ suc 𝐴
19 fnsnfv 6981 . . . . . . . . . . 11 ((𝑓 Fn suc 𝐴𝐴 ∈ suc 𝐴) → {(𝑓𝐴)} = (𝑓 “ {𝐴}))
2016, 18, 19sylancl 584 . . . . . . . . . 10 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → {(𝑓𝐴)} = (𝑓 “ {𝐴}))
2120difeq2d 4121 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → ((𝑓 “ suc 𝐴) ∖ {(𝑓𝐴)}) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
22 imadmrn 6079 . . . . . . . . . . . 12 (𝑓 “ dom 𝑓) = ran 𝑓
2322eqcomi 2735 . . . . . . . . . . 11 ran 𝑓 = (𝑓 “ dom 𝑓)
24 f1ofo 6850 . . . . . . . . . . . 12 (𝑓:suc 𝐴1-1-onto→suc 𝐵𝑓:suc 𝐴onto→suc 𝐵)
25 forn 6818 . . . . . . . . . . . 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 6847 . . . . . . . . . . . 12 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → dom 𝑓 = suc 𝐴)
2827imaeq2d 6069 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓 “ dom 𝑓) = (𝑓 “ suc 𝐴))
2923, 26, 283eqtr3a 2790 . . . . . . . . . 10 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → suc 𝐵 = (𝑓 “ suc 𝐴))
3029difeq1d 4120 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (suc 𝐵 ∖ {(𝑓𝐴)}) = ((𝑓 “ suc 𝐴) ∖ {(𝑓𝐴)}))
31 dff1o3 6849 . . . . . . . . . 10 (𝑓:suc 𝐴1-1-onto→suc 𝐵 ↔ (𝑓:suc 𝐴onto→suc 𝐵 ∧ Fun 𝑓))
32 imadif 6643 . . . . . . . . . 10 (Fun 𝑓 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
3331, 32simplbiim 503 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
3421, 30, 333eqtr4rd 2777 . . . . . . . 8 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = (suc 𝐵 ∖ {(𝑓𝐴)}))
3515, 34sylan9eq 2786 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (𝑓𝐴) = (suc 𝐵 ∖ {(𝑓𝐴)}))
3611, 35breqtrd 5179 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}))
37 fnfvelrn 7094 . . . . . . . . . . . 12 ((𝑓 Fn suc 𝐴𝐴 ∈ suc 𝐴) → (𝑓𝐴) ∈ ran 𝑓)
3816, 18, 37sylancl 584 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓𝐴) ∈ ran 𝑓)
3925eleq2d 2812 . . . . . . . . . . . 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 9241 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ (𝑓𝐴) ∈ suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}))
4341, 42sylan2 591 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}))
44 nnfi 9205 . . . . . . . . . . 11 (𝐵 ∈ ω → 𝐵 ∈ Fin)
45 ensymfib 9221 . . . . . . . . . . 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 479 . . . . . . . . 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 9222 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵) → 𝐴𝐵)
503, 49syl3an1 1160 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵) → 𝐴𝐵)
5148, 50syl3an3 1162 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵)) → 𝐴𝐵)
52513expa 1115 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)})) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵)) → 𝐴𝐵)
5336, 52syldanl 600 . . . . 5 (((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵)) → 𝐴𝐵)
5453anandirs 677 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → 𝐴𝐵)
5554ex 411 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑓:suc 𝐴1-1-onto→suc 𝐵𝐴𝐵))
5655exlimdv 1929 . 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 394   = wceq 1534  wex 1774  wcel 2099  Vcvv 3462  cdif 3944  wss 3947  {csn 4633   class class class wbr 5153  ccnv 5681  dom cdm 5682  ran crn 5683  cima 5685  Ord word 6375  suc csuc 6378  Fun wfun 6548   Fn wfn 6549  1-1wf1 6551  ontowfo 6552  1-1-ontowf1o 6553  cfv 6554  ωcom 7876  cen 8971  Fincfn 8974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-om 7877  df-1o 8496  df-en 8975  df-fin 8978
This theorem is referenced by:  nneneq  9243
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