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 𝐵 → 𝐴 ≈ 𝐵))

Proof of Theorem phplem2

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 8984	. 2 ⊢ (suc 𝐴 ≈ suc 𝐵 ↔ ∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵)
2		f1of1 6842	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–1-1→suc 𝐵)
3		nnfi 9205	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
4		sssucid 6456	. . . . . . . . . 10 ⊢ 𝐴 ⊆ suc 𝐴
5		f1imaenfi 9232	. . . . . . . . . 10 ⊢ ((𝑓:suc 𝐴–1-1→suc 𝐵 ∧ 𝐴 ⊆ suc 𝐴 ∧ 𝐴 ∈ Fin) → (𝑓 “ 𝐴) ≈ 𝐴)
6	4, 5	mp3an2 1446	. . . . . . . . 9 ⊢ ((𝑓:suc 𝐴–1-1→suc 𝐵 ∧ 𝐴 ∈ Fin) → (𝑓 “ 𝐴) ≈ 𝐴)
7	2, 3, 6	syl2anr 595	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) ≈ 𝐴)
8		ensymfib 9221	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (𝐴 ≈ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐴) ≈ 𝐴))
9	3, 8	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴 ≈ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐴) ≈ 𝐴))
10	9	adantr 479	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝐴 ≈ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐴) ≈ 𝐴))
11	7, 10	mpbird 256	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (𝑓 “ 𝐴))
12		nnord 7884	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → Ord 𝐴)
13		orddif 6472	. . . . . . . . . 10 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴}))
15	14	imaeq2d 6069	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝑓 “ 𝐴) = (𝑓 “ (suc 𝐴 ∖ {𝐴})))
16		f1ofn 6844	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓 Fn suc 𝐴)
17		phplem2.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ V
18	17	sucid 6458	. . . . . . . . . . 11 ⊢ 𝐴 ∈ suc 𝐴
19		fnsnfv 6981	. . . . . . . . . . 11 ⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
20	16, 18, 19	sylancl 584	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
21	20	difeq2d 4121	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
22		imadmrn 6079	. . . . . . . . . . . 12 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
23	22	eqcomi 2735	. . . . . . . . . . 11 ⊢ ran 𝑓 = (𝑓 “ dom 𝑓)
24		f1ofo 6850	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–onto→suc 𝐵)
25		forn 6818	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–onto→suc 𝐵 → ran 𝑓 = suc 𝐵)
26	24, 25	syl 17	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → ran 𝑓 = suc 𝐵)
27		f1odm 6847	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → dom 𝑓 = suc 𝐴)
28	27	imaeq2d 6069	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ dom 𝑓) = (𝑓 “ suc 𝐴))
29	23, 26, 28	3eqtr3a 2790	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → suc 𝐵 = (𝑓 “ suc 𝐴))
30	29	difeq1d 4120	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (suc 𝐵 ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)}))
31		dff1o3 6849	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 ↔ (𝑓:suc 𝐴–onto→suc 𝐵 ∧ Fun ◡𝑓))
32		imadif 6643	. . . . . . . . . 10 ⊢ (Fun ◡𝑓 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
33	31, 32	simplbiim 503	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
34	21, 30, 33	3eqtr4rd 2777	. . . . . . . 8 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = (suc 𝐵 ∖ {(𝑓‘𝐴)}))
35	15, 34	sylan9eq 2786	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) = (suc 𝐵 ∖ {(𝑓‘𝐴)}))
36	11, 35	breqtrd 5179	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
37		fnfvelrn 7094	. . . . . . . . . . . 12 ⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → (𝑓‘𝐴) ∈ ran 𝑓)
38	16, 18, 37	sylancl 584	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓‘𝐴) ∈ ran 𝑓)
39	25	eleq2d 2812	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–onto→suc 𝐵 → ((𝑓‘𝐴) ∈ ran 𝑓 ↔ (𝑓‘𝐴) ∈ suc 𝐵))
40	24, 39	syl 17	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → ((𝑓‘𝐴) ∈ ran 𝑓 ↔ (𝑓‘𝐴) ∈ suc 𝐵))
41	38, 40	mpbid 231	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓‘𝐴) ∈ suc 𝐵)
42		phplem1 9241	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ω ∧ (𝑓‘𝐴) ∈ suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
43	41, 42	sylan2 591	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
44		nnfi 9205	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ω → 𝐵 ∈ Fin)
45		ensymfib 9221	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → (𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ↔ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵))
46	44, 45	syl 17	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ↔ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵))
47	46	adantr 479	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ↔ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵))
48	43, 47	mpbid 231	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵)
49		entrfil 9222	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵) → 𝐴 ≈ 𝐵)
50	3, 49	syl3an1 1160	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵) → 𝐴 ≈ 𝐵)
51	48, 50	syl3an3 1162	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵)) → 𝐴 ≈ 𝐵)
52	51	3expa 1115	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)})) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵)) → 𝐴 ≈ 𝐵)
53	36, 52	syldanl 600	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵)) → 𝐴 ≈ 𝐵)
54	53	anandirs 677	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ 𝐵)
55	54	ex 411	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝐴 ≈ 𝐵))
56	55	exlimdv 1929	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝐴 ≈ 𝐵))
57	1, 56	biimtrid 241	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))

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  –1-1→wf1 6551  –onto→wfo 6552  –1-1-onto→wf1o 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