Theorem dif1en 8907

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. For a proof with fewer symbols using ax-pow 5283, see dif1enALT 8980. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5283. (Revised by BTernaryTau, 26-Aug-2024.)

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		bren 8701	. . 3 ⊢ (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀)
2		19.41v 1954	. . . . 5 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω)) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω)))
3		3anass 1093	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) ↔ (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω)))
4	3	exbii 1851	. . . . 5 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) ↔ ∃𝑓(𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω)))
5		3anass 1093	. . . . 5 ⊢ ((∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω)))
6	2, 4, 5	3bitr4i 302	. . . 4 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω))
7		sucidg 6329	. . . . . . . . 9 ⊢ (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀)
8		f1ocnvdm 7137	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
9	8	3adant2 1129	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
10		f1ofvswap 7158	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ (◡𝑓‘𝑀) ∈ 𝐴) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀)
11	9, 10	syld3an3 1407	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ suc 𝑀) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀)
12		f1ocnvfv2 7130	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝑓‘(◡𝑓‘𝑀)) = 𝑀)
13	12	opeq2d 4808	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩ = ⟨𝑋, 𝑀⟩)
14	13	preq1d 4672	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → {⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩} = {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})
15	14	uneq2d 4093	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}) = ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}))
16	15	f1oeq1d 6695	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀 ↔ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀))
17	16	3adant2 1129	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ suc 𝑀) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀 ↔ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀))
18	11, 17	mpbid 231	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ suc 𝑀) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀)
19		f1ofun 6702	. . . . . . . . . . 11 ⊢ (((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀 → Fun ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}))
20		opex 5373	. . . . . . . . . . . . . 14 ⊢ ⟨𝑋, 𝑀⟩ ∈ V
21	20	prid1 4695	. . . . . . . . . . . . 13 ⊢ ⟨𝑋, 𝑀⟩ ∈ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}
22		elun2 4107	. . . . . . . . . . . . 13 ⊢ (⟨𝑋, 𝑀⟩ ∈ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩} → ⟨𝑋, 𝑀⟩ ∈ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}))
23	21, 22	ax-mp 5	. . . . . . . . . . . 12 ⊢ ⟨𝑋, 𝑀⟩ ∈ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})
24		funopfv 6803	. . . . . . . . . . . 12 ⊢ (Fun ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}) → (⟨𝑋, 𝑀⟩ ∈ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑋) = 𝑀))
25	23, 24	mpi 20	. . . . . . . . . . 11 ⊢ (Fun ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑋) = 𝑀)
26	18, 19, 25	3syl 18	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ suc 𝑀) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑋) = 𝑀)
27		simp2 1135	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ suc 𝑀) → 𝑋 ∈ 𝐴)
28		f1ocnvfv 7131	. . . . . . . . . . 11 ⊢ ((((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴) → ((((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑋) = 𝑀 → (◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀) = 𝑋))
29	18, 27, 28	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ suc 𝑀) → ((((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑋) = 𝑀 → (◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀) = 𝑋))
30	26, 29	mpd 15	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ suc 𝑀) → (◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀) = 𝑋)
31	7, 30	syl3an3 1163	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → (◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀) = 𝑋)
32	31	sneqd 4570	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → {(◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀)} = {𝑋})
33	32	difeq2d 4053	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → (𝐴 ∖ {(◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀)}) = (𝐴 ∖ {𝑋}))
34		vex 3426	. . . . . . . . 9 ⊢ 𝑓 ∈ V
35	34	resex 5928	. . . . . . . 8 ⊢ (𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∈ V
36		prex 5350	. . . . . . . 8 ⊢ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩} ∈ V
37	35, 36	unex 7574	. . . . . . 7 ⊢ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}) ∈ V
38		simp3 1136	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → 𝑀 ∈ ω)
39	7, 18	syl3an3 1163	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀)
40		dif1enlem 8905	. . . . . . 7 ⊢ ((((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}) ∈ V ∧ 𝑀 ∈ ω ∧ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀)}) ≈ 𝑀)
41	37, 38, 39, 40	mp3an2i 1464	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → (𝐴 ∖ {(◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀)}) ≈ 𝑀)
42	33, 41	eqbrtrrd 5094	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
43	42	exlimiv 1934	. . . 4 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
44	6, 43	sylbir 234	. . 3 ⊢ ((∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
45	1, 44	syl3an1b 1401	. 2 ⊢ ((𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
46	45	3comr 1123	1 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881  {csn 4558  {cpr 4560  ⟨cop 4564   class class class wbr 5070  ^◡ccnv 5579   ↾ cres 5582  suc csuc 6253  Fun wfun 6412  –1-1-onto→wf1o 6417  ‘cfv 6418  ωcom 7687   ≈ cen 8688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-en 8692

This theorem is referenced by:  findcard  8908  enp1i  8982  findcard2OLD  8986  en2eleq  9695  en2other2  9696  mreexexlem4d  17273  f1otrspeq  18970  pmtrf  18978  pmtrmvd  18979  pmtrfinv  18984