Theorem dif1en 6337

Description: If a set 𝐴 is equinumerous to the successor of a natural number 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.)

Assertion

Ref	Expression
dif1en	⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Proof of Theorem dif1en

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 905	. . . 4 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → 𝐴 ≈ suc 𝑀)
2	1	ensymd 6263	. . 3 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → suc 𝑀 ≈ 𝐴)
3		bren 6228	. . 3 ⊢ (suc 𝑀 ≈ 𝐴 ↔ ∃𝑓 𝑓:suc 𝑀–1-1-onto→𝐴)
4	2, 3	sylib 127	. 2 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → ∃𝑓 𝑓:suc 𝑀–1-1-onto→𝐴)
5		peano2 4318	. . . . . . . 8 ⊢ (𝑀 ∈ ω → suc 𝑀 ∈ ω)
6		nnfi 6333	. . . . . . . 8 ⊢ (suc 𝑀 ∈ ω → suc 𝑀 ∈ Fin)
7	5, 6	syl 14	. . . . . . 7 ⊢ (𝑀 ∈ ω → suc 𝑀 ∈ Fin)
8	7	3ad2ant1 925	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → suc 𝑀 ∈ Fin)
9		enfii 6335	. . . . . 6 ⊢ ((suc 𝑀 ∈ Fin ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
10	8, 1, 9	syl2anc 391	. . . . 5 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → 𝐴 ∈ Fin)
11	10	adantr 261	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝐴 ∈ Fin)
12		simpl3 909	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑋 ∈ 𝐴)
13		f1of 5126	. . . . . 6 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → 𝑓:suc 𝑀⟶𝐴)
14	13	adantl 262	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑓:suc 𝑀⟶𝐴)
15		sucidg 4153	. . . . . . 7 ⊢ (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀)
16	15	3ad2ant1 925	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → 𝑀 ∈ suc 𝑀)
17	16	adantr 261	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 ∈ suc 𝑀)
18	14, 17	ffvelrnd 5303	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓‘𝑀) ∈ 𝐴)
19		fidifsnen 6331	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ (𝑓‘𝑀) ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
20	11, 12, 18, 19	syl3anc 1135	. . 3 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝐴 ∖ {𝑋}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
21		nnord 4334	. . . . . . . 8 ⊢ (𝑀 ∈ ω → Ord 𝑀)
22		orddif 4271	. . . . . . . 8 ⊢ (Ord 𝑀 → 𝑀 = (suc 𝑀 ∖ {𝑀}))
23	21, 22	syl 14	. . . . . . 7 ⊢ (𝑀 ∈ ω → 𝑀 = (suc 𝑀 ∖ {𝑀}))
24	23	3ad2ant1 925	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → 𝑀 = (suc 𝑀 ∖ {𝑀}))
25	24	adantr 261	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 = (suc 𝑀 ∖ {𝑀}))
26	23	eleq1d 2106	. . . . . . . . 9 ⊢ (𝑀 ∈ ω → (𝑀 ∈ ω ↔ (suc 𝑀 ∖ {𝑀}) ∈ ω))
27	26	ibi 165	. . . . . . . 8 ⊢ (𝑀 ∈ ω → (suc 𝑀 ∖ {𝑀}) ∈ ω)
28	27	3ad2ant1 925	. . . . . . 7 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (suc 𝑀 ∖ {𝑀}) ∈ ω)
29	28	adantr 261	. . . . . 6 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (suc 𝑀 ∖ {𝑀}) ∈ ω)
30		dff1o2 5131	. . . . . . . . 9 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 ↔ (𝑓 Fn suc 𝑀 ∧ Fun ◡𝑓 ∧ ran 𝑓 = 𝐴))
31	30	simp2bi 920	. . . . . . . 8 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → Fun ◡𝑓)
32	31	adantl 262	. . . . . . 7 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → Fun ◡𝑓)
33		f1ofo 5133	. . . . . . . . 9 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → 𝑓:suc 𝑀–onto→𝐴)
34	33	adantl 262	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑓:suc 𝑀–onto→𝐴)
35		f1orel 5129	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → Rel 𝑓)
36	35	adantl 262	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → Rel 𝑓)
37		resdm 4649	. . . . . . . . . . 11 ⊢ (Rel 𝑓 → (𝑓 ↾ dom 𝑓) = 𝑓)
38	36, 37	syl 14	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ dom 𝑓) = 𝑓)
39		f1odm 5130	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → dom 𝑓 = suc 𝑀)
40	39	reseq2d 4612	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → (𝑓 ↾ dom 𝑓) = (𝑓 ↾ suc 𝑀))
41	40	adantl 262	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ dom 𝑓) = (𝑓 ↾ suc 𝑀))
42	38, 41	eqtr3d 2074	. . . . . . . . 9 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑓 = (𝑓 ↾ suc 𝑀))
43		foeq1 5102	. . . . . . . . 9 ⊢ (𝑓 = (𝑓 ↾ suc 𝑀) → (𝑓:suc 𝑀–onto→𝐴 ↔ (𝑓 ↾ suc 𝑀):suc 𝑀–onto→𝐴))
44	42, 43	syl 14	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓:suc 𝑀–onto→𝐴 ↔ (𝑓 ↾ suc 𝑀):suc 𝑀–onto→𝐴))
45	34, 44	mpbid 135	. . . . . . 7 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ suc 𝑀):suc 𝑀–onto→𝐴)
46		simpl1 907	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 ∈ ω)
47		f1osng 5167	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ω ∧ (𝑓‘𝑀) ∈ 𝐴) → {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–1-1-onto→{(𝑓‘𝑀)})
48	46, 18, 47	syl2anc 391	. . . . . . . . 9 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–1-1-onto→{(𝑓‘𝑀)})
49		f1ofo 5133	. . . . . . . . 9 ⊢ ({⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–1-1-onto→{(𝑓‘𝑀)} → {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–onto→{(𝑓‘𝑀)})
50	48, 49	syl 14	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–onto→{(𝑓‘𝑀)})
51		f1ofn 5127	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → 𝑓 Fn suc 𝑀)
52	51	adantl 262	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑓 Fn suc 𝑀)
53		fnressn 5349	. . . . . . . . . 10 ⊢ ((𝑓 Fn suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝑓 ↾ {𝑀}) = {⟨𝑀, (𝑓‘𝑀)⟩})
54	52, 17, 53	syl2anc 391	. . . . . . . . 9 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ {𝑀}) = {⟨𝑀, (𝑓‘𝑀)⟩})
55		foeq1 5102	. . . . . . . . 9 ⊢ ((𝑓 ↾ {𝑀}) = {⟨𝑀, (𝑓‘𝑀)⟩} → ((𝑓 ↾ {𝑀}):{𝑀}–onto→{(𝑓‘𝑀)} ↔ {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–onto→{(𝑓‘𝑀)}))
56	54, 55	syl 14	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → ((𝑓 ↾ {𝑀}):{𝑀}–onto→{(𝑓‘𝑀)} ↔ {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–onto→{(𝑓‘𝑀)}))
57	50, 56	mpbird 156	. . . . . . 7 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ {𝑀}):{𝑀}–onto→{(𝑓‘𝑀)})
58		resdif 5148	. . . . . . 7 ⊢ ((Fun ◡𝑓 ∧ (𝑓 ↾ suc 𝑀):suc 𝑀–onto→𝐴 ∧ (𝑓 ↾ {𝑀}):{𝑀}–onto→{(𝑓‘𝑀)}) → (𝑓 ↾ (suc 𝑀 ∖ {𝑀})):(suc 𝑀 ∖ {𝑀})–1-1-onto→(𝐴 ∖ {(𝑓‘𝑀)}))
59	32, 45, 57, 58	syl3anc 1135	. . . . . 6 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ (suc 𝑀 ∖ {𝑀})):(suc 𝑀 ∖ {𝑀})–1-1-onto→(𝐴 ∖ {(𝑓‘𝑀)}))
60		f1oeng 6237	. . . . . 6 ⊢ (((suc 𝑀 ∖ {𝑀}) ∈ ω ∧ (𝑓 ↾ (suc 𝑀 ∖ {𝑀})):(suc 𝑀 ∖ {𝑀})–1-1-onto→(𝐴 ∖ {(𝑓‘𝑀)})) → (suc 𝑀 ∖ {𝑀}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
61	29, 59, 60	syl2anc 391	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (suc 𝑀 ∖ {𝑀}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
62	25, 61	eqbrtrd 3784	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
63	62	ensymd 6263	. . 3 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝐴 ∖ {(𝑓‘𝑀)}) ≈ 𝑀)
64		entr 6264	. . 3 ⊢ (((𝐴 ∖ {𝑋}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}) ∧ (𝐴 ∖ {(𝑓‘𝑀)}) ≈ 𝑀) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
65	20, 63, 64	syl2anc 391	. 2 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
66	4, 65	exlimddv 1778	1 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243  ∃wex 1381   ∈ wcel 1393   ∖ cdif 2914  {csn 3375  ⟨cop 3378   class class class wbr 3764  Ord word 4099  suc csuc 4102  ωcom 4313  ^◡ccnv 4344  dom cdm 4345  ran crn 4346   ↾ cres 4347  Rel wrel 4350  Fun wfun 4896   Fn wfn 4897  ⟶wf 4898  –onto→wfo 4900  –1-1-onto→wf1o 4901  ‘cfv 4902   ≈ cen 6219  Fincfn 6221

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-er 6106  df-en 6222  df-fin 6224

This theorem is referenced by:  findcard  6345  findcard2  6346  findcard2s  6347  diffisn  6350