Theorem phplem4 6821

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.)

Hypotheses

Ref	Expression
phplem2.1	⊢ 𝐴 ∈ V
phplem2.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
phplem4	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))

Proof of Theorem phplem4

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 6713	. 2 ⊢ (suc 𝐴 ≈ suc 𝐵 ↔ ∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵)
2		f1of1 5431	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–1-1→suc 𝐵)
3	2	adantl 275	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝑓:suc 𝐴–1-1→suc 𝐵)
4		phplem2.2	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
5	4	sucex 4476	. . . . . . . . 9 ⊢ suc 𝐵 ∈ V
6		sssucid 4393	. . . . . . . . . 10 ⊢ 𝐴 ⊆ suc 𝐴
7		phplem2.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
8		f1imaen2g 6759	. . . . . . . . . 10 ⊢ (((𝑓:suc 𝐴–1-1→suc 𝐵 ∧ suc 𝐵 ∈ V) ∧ (𝐴 ⊆ suc 𝐴 ∧ 𝐴 ∈ V)) → (𝑓 “ 𝐴) ≈ 𝐴)
9	6, 7, 8	mpanr12 436	. . . . . . . . 9 ⊢ ((𝑓:suc 𝐴–1-1→suc 𝐵 ∧ suc 𝐵 ∈ V) → (𝑓 “ 𝐴) ≈ 𝐴)
10	3, 5, 9	sylancl 410	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) ≈ 𝐴)
11	10	ensymd 6749	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (𝑓 “ 𝐴))
12		nnord 4589	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → Ord 𝐴)
13		orddif 4524	. . . . . . . . . 10 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))
14	12, 13	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴}))
15	14	imaeq2d 4946	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝑓 “ 𝐴) = (𝑓 “ (suc 𝐴 ∖ {𝐴})))
16		f1ofn 5433	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓 Fn suc 𝐴)
17	7	sucid 4395	. . . . . . . . . . 11 ⊢ 𝐴 ∈ suc 𝐴
18		fnsnfv 5545	. . . . . . . . . . 11 ⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
19	16, 17, 18	sylancl 410	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
20	19	difeq2d 3240	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
21		imadmrn 4956	. . . . . . . . . . . 12 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
22	21	eqcomi 2169	. . . . . . . . . . 11 ⊢ ran 𝑓 = (𝑓 “ dom 𝑓)
23		f1ofo 5439	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–onto→suc 𝐵)
24		forn 5413	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–onto→suc 𝐵 → ran 𝑓 = suc 𝐵)
25	23, 24	syl 14	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → ran 𝑓 = suc 𝐵)
26		f1odm 5436	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → dom 𝑓 = suc 𝐴)
27	26	imaeq2d 4946	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ dom 𝑓) = (𝑓 “ suc 𝐴))
28	22, 25, 27	3eqtr3a 2223	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → suc 𝐵 = (𝑓 “ suc 𝐴))
29	28	difeq1d 3239	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (suc 𝐵 ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)}))
30		dff1o3 5438	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 ↔ (𝑓:suc 𝐴–onto→suc 𝐵 ∧ Fun ◡𝑓))
31	30	simprbi 273	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → Fun ◡𝑓)
32		imadif 5268	. . . . . . . . . 10 ⊢ (Fun ◡𝑓 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
33	31, 32	syl 14	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
34	20, 29, 33	3eqtr4rd 2209	. . . . . . . 8 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = (suc 𝐵 ∖ {(𝑓‘𝐴)}))
35	15, 34	sylan9eq 2219	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) = (suc 𝐵 ∖ {(𝑓‘𝐴)}))
36	11, 35	breqtrd 4008	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
37		fnfvelrn 5617	. . . . . . . . . 10 ⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → (𝑓‘𝐴) ∈ ran 𝑓)
38	16, 17, 37	sylancl 410	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓‘𝐴) ∈ ran 𝑓)
39	24	eleq2d 2236	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–onto→suc 𝐵 → ((𝑓‘𝐴) ∈ ran 𝑓 ↔ (𝑓‘𝐴) ∈ suc 𝐵))
40	23, 39	syl 14	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → ((𝑓‘𝐴) ∈ ran 𝑓 ↔ (𝑓‘𝐴) ∈ suc 𝐵))
41	38, 40	mpbid 146	. . . . . . . 8 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓‘𝐴) ∈ suc 𝐵)
42		vex 2729	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
43	42, 7	fvex 5506	. . . . . . . . 9 ⊢ (𝑓‘𝐴) ∈ V
44	4, 43	phplem3 6820	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ (𝑓‘𝐴) ∈ suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
45	41, 44	sylan2 284	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
46	45	ensymd 6749	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵)
47		entr 6750	. . . . . 6 ⊢ ((𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵) → 𝐴 ≈ 𝐵)
48	36, 46, 47	syl2an 287	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵)) → 𝐴 ≈ 𝐵)
49	48	anandirs 583	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ 𝐵)
50	49	ex 114	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝐴 ≈ 𝐵))
51	50	exlimdv 1807	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝐴 ≈ 𝐵))
52	1, 51	syl5bi 151	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2726   ∖ cdif 3113   ⊆ wss 3116  {csn 3576   class class class wbr 3982  Ord word 4340  suc csuc 4343  ωcom 4567  ^◡ccnv 4603  dom cdm 4604  ran crn 4605   “ cima 4607  Fun wfun 5182   Fn wfn 5183  –1-1→wf1 5185  –onto→wfo 5186  –1-1-onto→wf1o 5187  ‘cfv 5188   ≈ cen 6704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-er 6501  df-en 6707

This theorem is referenced by:  nneneq  6823  php5  6824