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 𝐵 → 𝐴 ≈ 𝐵))

Proof of Theorem phplem2

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 8995	. 2 ⊢ (suc 𝐴 ≈ suc 𝐵 ↔ ∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵)
2		f1of1 6847	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–1-1→suc 𝐵)
3		nnfi 9207	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
4		sssucid 6464	. . . . . . . . . 10 ⊢ 𝐴 ⊆ suc 𝐴
5		f1imaenfi 9235	. . . . . . . . . 10 ⊢ ((𝑓:suc 𝐴–1-1→suc 𝐵 ∧ 𝐴 ⊆ suc 𝐴 ∧ 𝐴 ∈ Fin) → (𝑓 “ 𝐴) ≈ 𝐴)
6	4, 5	mp3an2 1451	. . . . . . . . 9 ⊢ ((𝑓:suc 𝐴–1-1→suc 𝐵 ∧ 𝐴 ∈ Fin) → (𝑓 “ 𝐴) ≈ 𝐴)
7	2, 3, 6	syl2anr 597	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) ≈ 𝐴)
8		ensymfib 9224	. . . . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝐴 ≈ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐴) ≈ 𝐴))
11	7, 10	mpbird 257	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (𝑓 “ 𝐴))
12		nnord 7895	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → Ord 𝐴)
13		orddif 6480	. . . . . . . . . 10 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴}))
15	14	imaeq2d 6078	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝑓 “ 𝐴) = (𝑓 “ (suc 𝐴 ∖ {𝐴})))
16		f1ofn 6849	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓 Fn suc 𝐴)
17		phplem2.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ V
18	17	sucid 6466	. . . . . . . . . . 11 ⊢ 𝐴 ∈ suc 𝐴
19		fnsnfv 6988	. . . . . . . . . . 11 ⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
20	16, 18, 19	sylancl 586	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
21	20	difeq2d 4126	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
22		imadmrn 6088	. . . . . . . . . . . 12 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
23	22	eqcomi 2746	. . . . . . . . . . 11 ⊢ ran 𝑓 = (𝑓 “ dom 𝑓)
24		f1ofo 6855	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–onto→suc 𝐵)
25		forn 6823	. . . . . . . . . . . 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 6852	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → dom 𝑓 = suc 𝐴)
28	27	imaeq2d 6078	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ dom 𝑓) = (𝑓 “ suc 𝐴))
29	23, 26, 28	3eqtr3a 2801	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → suc 𝐵 = (𝑓 “ suc 𝐴))
30	29	difeq1d 4125	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (suc 𝐵 ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)}))
31		dff1o3 6854	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 ↔ (𝑓:suc 𝐴–onto→suc 𝐵 ∧ Fun ◡𝑓))
32		imadif 6650	. . . . . . . . . 10 ⊢ (Fun ◡𝑓 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
33	31, 32	simplbiim 504	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
34	21, 30, 33	3eqtr4rd 2788	. . . . . . . 8 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = (suc 𝐵 ∖ {(𝑓‘𝐴)}))
35	15, 34	sylan9eq 2797	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) = (suc 𝐵 ∖ {(𝑓‘𝐴)}))
36	11, 35	breqtrd 5169	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
37		fnfvelrn 7100	. . . . . . . . . . . 12 ⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → (𝑓‘𝐴) ∈ ran 𝑓)
38	16, 18, 37	sylancl 586	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓‘𝐴) ∈ ran 𝑓)
39	25	eleq2d 2827	. . . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓‘𝐴) ∈ suc 𝐵)
42		phplem1 9244	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ω ∧ (𝑓‘𝐴) ∈ suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
43	41, 42	sylan2 593	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
44		nnfi 9207	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ω → 𝐵 ∈ Fin)
45		ensymfib 9224	. . . . . . . . . . 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 480	. . . . . . . . 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 232	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵)
49		entrfil 9225	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵) → 𝐴 ≈ 𝐵)
50	3, 49	syl3an1 1164	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵) → 𝐴 ≈ 𝐵)
51	48, 50	syl3an3 1166	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵)) → 𝐴 ≈ 𝐵)
52	51	3expa 1119	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)})) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵)) → 𝐴 ≈ 𝐵)
53	36, 52	syldanl 602	. . . . 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 679	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ 𝐵)
55	54	ex 412	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝐴 ≈ 𝐵))
56	55	exlimdv 1933	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝐴 ≈ 𝐵))
57	1, 56	biimtrid 242	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))

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  –1-1→wf1 6558  –onto→wfo 6559  –1-1-onto→wf1o 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