Theorem dif1enOLD 9103

Description: Obsolete version of dif1en 9101 as of 6-Jan-2025. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5315. (Revised by BTernaryTau, 26-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dif1enOLD	⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Proof of Theorem dif1enOLD

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 8905	. . 3 ⊢ (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀)
2		19.41v 1949	. . . . 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 1094	. . . . . 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 1848	. . . . 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 1094	. . . . 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 303	. . . 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 6403	. . . . . . . . 9 ⊢ (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀)
8		f1ocnvdm 7242	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
9	8	3adant2 1131	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
10		f1ofvswap 7263	. . . . . . . . . . . . 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 1411	. . . . . . . . . . . 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 7234	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝑓‘(◡𝑓‘𝑀)) = 𝑀)
13	12	opeq2d 4840	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩ = ⟨𝑋, 𝑀⟩)
14	13	preq1d 4699	. . . . . . . . . . . . . . 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 4127	. . . . . . . . . . . . . 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 6777	. . . . . . . . . . . . 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 1131	. . . . . . . . . . . 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 232	. . . . . . . . . . 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 6784	. . . . . . . . . . 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 5419	. . . . . . . . . . . . . 14 ⊢ ⟨𝑋, 𝑀⟩ ∈ V
21	20	prid1 4722	. . . . . . . . . . . . 13 ⊢ ⟨𝑋, 𝑀⟩ ∈ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}
22		elun2 4142	. . . . . . . . . . . . 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 6892	. . . . . . . . . . . 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 1137	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ suc 𝑀) → 𝑋 ∈ 𝐴)
28		f1ocnvfv 7235	. . . . . . . . . . 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 584	. . . . . . . . . 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 1165	. . . . . . . 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 4597	. . . . . . 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 4085	. . . . . 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 3448	. . . . . . . . 9 ⊢ 𝑓 ∈ V
35	34	resex 5989	. . . . . . . 8 ⊢ (𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∈ V
36		prex 5387	. . . . . . . 8 ⊢ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩} ∈ V
37	35, 36	unex 7700	. . . . . . 7 ⊢ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}) ∈ V
38		simp3 1138	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → 𝑀 ∈ ω)
39	7, 18	syl3an3 1165	. . . . . . 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		dif1enlemOLD 9098	. . . . . . 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 1468	. . . . . 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 5126	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
43	42	exlimiv 1930	. . . 4 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
44	6, 43	sylbir 235	. . 3 ⊢ ((∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
45	1, 44	syl3an1b 1405	. 2 ⊢ ((𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
46	45	3comr 1125	1 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3444   ∖ cdif 3908   ∪ cun 3909  {csn 4585  {cpr 4587  ⟨cop 4591   class class class wbr 5102  ^◡ccnv 5630   ↾ cres 5633  suc csuc 6322  Fun wfun 6493  –1-1-onto→wf1o 6498  ‘cfv 6499  ωcom 7822   ≈ cen 8892

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-en 8896

This theorem is referenced by: (None)