Theorem dif1enOLD 9228

Description: Obsolete version of dif1en 9226 as of 6-Jan-2025. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5383. (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 9013	. . 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 1095	. . . . . 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 1846	. . . . 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 1095	. . . . 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 6476	. . . . . . . . 9 ⊢ (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀)
8		f1ocnvdm 7321	. . . . . . . . . . . . . 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 7342	. . . . . . . . . . . . 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 1409	. . . . . . . . . . . 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 7313	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝑓‘(◡𝑓‘𝑀)) = 𝑀)
13	12	opeq2d 4904	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩ = ⟨𝑋, 𝑀⟩)
14	13	preq1d 4764	. . . . . . . . . . . . . . 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 4191	. . . . . . . . . . . . . 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 6857	. . . . . . . . . . . . 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 6864	. . . . . . . . . . 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 5484	. . . . . . . . . . . . . 14 ⊢ ⟨𝑋, 𝑀⟩ ∈ V
21	20	prid1 4787	. . . . . . . . . . . . 13 ⊢ ⟨𝑋, 𝑀⟩ ∈ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}
22		elun2 4206	. . . . . . . . . . . . 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 6972	. . . . . . . . . . . 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 7314	. . . . . . . . . . 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 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 4660	. . . . . . 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 4149	. . . . . 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 3492	. . . . . . . . 9 ⊢ 𝑓 ∈ V
35	34	resex 6058	. . . . . . . 8 ⊢ (𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∈ V
36		prex 5452	. . . . . . . 8 ⊢ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩} ∈ V
37	35, 36	unex 7779	. . . . . . 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 9223	. . . . . . 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 1466	. . . . . 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 5190	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ ω) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
43	42	exlimiv 1929	. . . 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 1403	. 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 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973   ∪ cun 3974  {csn 4648  {cpr 4650  ⟨cop 4654   class class class wbr 5166  ^◡ccnv 5699   ↾ cres 5702  suc csuc 6397  Fun wfun 6567  –1-1-onto→wf1o 6572  ‘cfv 6573  ωcom 7903   ≈ cen 9000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-en 9004

This theorem is referenced by: (None)