Theorem ssenen 7152

Description: Equinumerosity of equinumerous subsets of a set. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
ssenen	⊢ (𝐴 ≈ 𝐵 → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐶)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝑥 ≈ 𝐶)})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem ssenen

Dummy variables 𝑦 𝑧 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 7030	. . 3 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2		f1odm 5643	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → dom 𝑓 = 𝐴)
3		vex 2824	. . . . . . . 8 ⊢ 𝑓 ∈ V
4	3	dmex 5049	. . . . . . 7 ⊢ dom 𝑓 ∈ V
5	2, 4	eqeltrrdi 2330	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐴 ∈ V)
6		pwexg 4317	. . . . . 6 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
7		inex1g 4269	. . . . . 6 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ∈ V)
8	5, 6, 7	3syl 17	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ∈ V)
9		f1ofo 5646	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
10		forn 5618	. . . . . . . 8 ⊢ (𝑓:𝐴–onto→𝐵 → ran 𝑓 = 𝐵)
11	9, 10	syl 14	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ran 𝑓 = 𝐵)
12	3	rnex 5050	. . . . . . 7 ⊢ ran 𝑓 ∈ V
13	11, 12	eqeltrrdi 2330	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐵 ∈ V)
14		pwexg 4317	. . . . . 6 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
15		inex1g 4269	. . . . . 6 ⊢ (𝒫 𝐵 ∈ V → (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ∈ V)
16	13, 14, 15	3syl 17	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ∈ V)
17		f1of1 5638	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–1-1→𝐵)
18	17	adantr 276	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) → 𝑓:𝐴–1-1→𝐵)
19	13	adantr 276	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) → 𝐵 ∈ V)
20		simpr 110	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
21		vex 2824	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	21	a1i 9	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ V)
23		f1imaen2g 7080	. . . . . . . . . 10 ⊢ (((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑦 ∈ V)) → (𝑓 “ 𝑦) ≈ 𝑦)
24	18, 19, 20, 22, 23	syl22anc 1279	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) → (𝑓 “ 𝑦) ≈ 𝑦)
25		entr 7071	. . . . . . . . 9 ⊢ (((𝑓 “ 𝑦) ≈ 𝑦 ∧ 𝑦 ≈ 𝐶) → (𝑓 “ 𝑦) ≈ 𝐶)
26	24, 25	sylan 283	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑦 ≈ 𝐶) → (𝑓 “ 𝑦) ≈ 𝐶)
27	26	expl 378	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ((𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝐶) → (𝑓 “ 𝑦) ≈ 𝐶))
28		imassrn 5137	. . . . . . . . 9 ⊢ (𝑓 “ 𝑦) ⊆ ran 𝑓
29	28, 10	sseqtrid 3298	. . . . . . . 8 ⊢ (𝑓:𝐴–onto→𝐵 → (𝑓 “ 𝑦) ⊆ 𝐵)
30	9, 29	syl 14	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑓 “ 𝑦) ⊆ 𝐵)
31	27, 30	jctild 316	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ((𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝐶) → ((𝑓 “ 𝑦) ⊆ 𝐵 ∧ (𝑓 “ 𝑦) ≈ 𝐶)))
32		elin 3412	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}))
33	21	elpw 3694	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
34		breq1 4133	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝐶 ↔ 𝑦 ≈ 𝐶))
35	21, 34	elab 2970	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶} ↔ 𝑦 ≈ 𝐶)
36	33, 35	anbi12i 464	. . . . . . 7 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝐶))
37	32, 36	bitri 184	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝐶))
38		elin 3412	. . . . . . 7 ⊢ ((𝑓 “ 𝑦) ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ ((𝑓 “ 𝑦) ∈ 𝒫 𝐵 ∧ (𝑓 “ 𝑦) ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}))
39	3	imaex 5141	. . . . . . . . 9 ⊢ (𝑓 “ 𝑦) ∈ V
40	39	elpw 3694	. . . . . . . 8 ⊢ ((𝑓 “ 𝑦) ∈ 𝒫 𝐵 ↔ (𝑓 “ 𝑦) ⊆ 𝐵)
41		breq1 4133	. . . . . . . . 9 ⊢ (𝑥 = (𝑓 “ 𝑦) → (𝑥 ≈ 𝐶 ↔ (𝑓 “ 𝑦) ≈ 𝐶))
42	39, 41	elab 2970	. . . . . . . 8 ⊢ ((𝑓 “ 𝑦) ∈ {𝑥 ∣ 𝑥 ≈ 𝐶} ↔ (𝑓 “ 𝑦) ≈ 𝐶)
43	40, 42	anbi12i 464	. . . . . . 7 ⊢ (((𝑓 “ 𝑦) ∈ 𝒫 𝐵 ∧ (𝑓 “ 𝑦) ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ ((𝑓 “ 𝑦) ⊆ 𝐵 ∧ (𝑓 “ 𝑦) ≈ 𝐶))
44	38, 43	bitri 184	. . . . . 6 ⊢ ((𝑓 “ 𝑦) ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ ((𝑓 “ 𝑦) ⊆ 𝐵 ∧ (𝑓 “ 𝑦) ≈ 𝐶))
45	31, 37, 44	3imtr4g 205	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) → (𝑓 “ 𝑦) ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶})))
46		f1ocnv 5652	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1-onto→𝐴)
47		f1of1 5638	. . . . . . . . . . . 12 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → ◡𝑓:𝐵–1-1→𝐴)
48		f1f1orn 5650	. . . . . . . . . . . 12 ⊢ (◡𝑓:𝐵–1-1→𝐴 → ◡𝑓:𝐵–1-1-onto→ran ◡𝑓)
49		f1of1 5638	. . . . . . . . . . . 12 ⊢ (◡𝑓:𝐵–1-1-onto→ran ◡𝑓 → ◡𝑓:𝐵–1-1→ran ◡𝑓)
50	47, 48, 49	3syl 17	. . . . . . . . . . 11 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → ◡𝑓:𝐵–1-1→ran ◡𝑓)
51		vex 2824	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
52	51	f1imaen 7081	. . . . . . . . . . 11 ⊢ ((◡𝑓:𝐵–1-1→ran ◡𝑓 ∧ 𝑧 ⊆ 𝐵) → (◡𝑓 “ 𝑧) ≈ 𝑧)
53	50, 52	sylan 283	. . . . . . . . . 10 ⊢ ((◡𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑧 ⊆ 𝐵) → (◡𝑓 “ 𝑧) ≈ 𝑧)
54		entr 7071	. . . . . . . . . 10 ⊢ (((◡𝑓 “ 𝑧) ≈ 𝑧 ∧ 𝑧 ≈ 𝐶) → (◡𝑓 “ 𝑧) ≈ 𝐶)
55	53, 54	sylan 283	. . . . . . . . 9 ⊢ (((◡𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑧 ≈ 𝐶) → (◡𝑓 “ 𝑧) ≈ 𝐶)
56	55	expl 378	. . . . . . . 8 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → ((𝑧 ⊆ 𝐵 ∧ 𝑧 ≈ 𝐶) → (◡𝑓 “ 𝑧) ≈ 𝐶))
57		f1ofo 5646	. . . . . . . . 9 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → ◡𝑓:𝐵–onto→𝐴)
58		imassrn 5137	. . . . . . . . . 10 ⊢ (◡𝑓 “ 𝑧) ⊆ ran ◡𝑓
59		forn 5618	. . . . . . . . . 10 ⊢ (◡𝑓:𝐵–onto→𝐴 → ran ◡𝑓 = 𝐴)
60	58, 59	sseqtrid 3298	. . . . . . . . 9 ⊢ (◡𝑓:𝐵–onto→𝐴 → (◡𝑓 “ 𝑧) ⊆ 𝐴)
61	57, 60	syl 14	. . . . . . . 8 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → (◡𝑓 “ 𝑧) ⊆ 𝐴)
62	56, 61	jctild 316	. . . . . . 7 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → ((𝑧 ⊆ 𝐵 ∧ 𝑧 ≈ 𝐶) → ((◡𝑓 “ 𝑧) ⊆ 𝐴 ∧ (◡𝑓 “ 𝑧) ≈ 𝐶)))
63	46, 62	syl 14	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ((𝑧 ⊆ 𝐵 ∧ 𝑧 ≈ 𝐶) → ((◡𝑓 “ 𝑧) ⊆ 𝐴 ∧ (◡𝑓 “ 𝑧) ≈ 𝐶)))
64		elin 3412	. . . . . . 7 ⊢ (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ (𝑧 ∈ 𝒫 𝐵 ∧ 𝑧 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}))
65	51	elpw 3694	. . . . . . . 8 ⊢ (𝑧 ∈ 𝒫 𝐵 ↔ 𝑧 ⊆ 𝐵)
66		breq1 4133	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ≈ 𝐶 ↔ 𝑧 ≈ 𝐶))
67	51, 66	elab 2970	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶} ↔ 𝑧 ≈ 𝐶)
68	65, 67	anbi12i 464	. . . . . . 7 ⊢ ((𝑧 ∈ 𝒫 𝐵 ∧ 𝑧 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ (𝑧 ⊆ 𝐵 ∧ 𝑧 ≈ 𝐶))
69	64, 68	bitri 184	. . . . . 6 ⊢ (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ (𝑧 ⊆ 𝐵 ∧ 𝑧 ≈ 𝐶))
70		elin 3412	. . . . . . 7 ⊢ ((◡𝑓 “ 𝑧) ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ ((◡𝑓 “ 𝑧) ∈ 𝒫 𝐴 ∧ (◡𝑓 “ 𝑧) ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}))
71	3	cnvex 5326	. . . . . . . . . 10 ⊢ ◡𝑓 ∈ V
72	71	imaex 5141	. . . . . . . . 9 ⊢ (◡𝑓 “ 𝑧) ∈ V
73	72	elpw 3694	. . . . . . . 8 ⊢ ((◡𝑓 “ 𝑧) ∈ 𝒫 𝐴 ↔ (◡𝑓 “ 𝑧) ⊆ 𝐴)
74		breq1 4133	. . . . . . . . 9 ⊢ (𝑥 = (◡𝑓 “ 𝑧) → (𝑥 ≈ 𝐶 ↔ (◡𝑓 “ 𝑧) ≈ 𝐶))
75	72, 74	elab 2970	. . . . . . . 8 ⊢ ((◡𝑓 “ 𝑧) ∈ {𝑥 ∣ 𝑥 ≈ 𝐶} ↔ (◡𝑓 “ 𝑧) ≈ 𝐶)
76	73, 75	anbi12i 464	. . . . . . 7 ⊢ (((◡𝑓 “ 𝑧) ∈ 𝒫 𝐴 ∧ (◡𝑓 “ 𝑧) ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ ((◡𝑓 “ 𝑧) ⊆ 𝐴 ∧ (◡𝑓 “ 𝑧) ≈ 𝐶))
77	70, 76	bitri 184	. . . . . 6 ⊢ ((◡𝑓 “ 𝑧) ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ ((◡𝑓 “ 𝑧) ⊆ 𝐴 ∧ (◡𝑓 “ 𝑧) ≈ 𝐶))
78	63, 69, 77	3imtr4g 205	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) → (◡𝑓 “ 𝑧) ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶})))
79		simpl 109	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝒫 𝐵 ∧ 𝑧 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) → 𝑧 ∈ 𝒫 𝐵)
80	79	elpwid 3700	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝒫 𝐵 ∧ 𝑧 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) → 𝑧 ⊆ 𝐵)
81	64, 80	sylbi 121	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) → 𝑧 ⊆ 𝐵)
82		imaeq2 5122	. . . . . . . . . . . 12 ⊢ (𝑦 = (◡𝑓 “ 𝑧) → (𝑓 “ 𝑦) = (𝑓 “ (◡𝑓 “ 𝑧)))
83		f1orel 5642	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → Rel 𝑓)
84		dfrel2 5238	. . . . . . . . . . . . . . . 16 ⊢ (Rel 𝑓 ↔ ◡◡𝑓 = 𝑓)
85	83, 84	sylib 122	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡◡𝑓 = 𝑓)
86	85	imaeq1d 5125	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (◡◡𝑓 “ (◡𝑓 “ 𝑧)) = (𝑓 “ (◡𝑓 “ 𝑧)))
87	86	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑧 ⊆ 𝐵) → (◡◡𝑓 “ (◡𝑓 “ 𝑧)) = (𝑓 “ (◡𝑓 “ 𝑧)))
88	46, 47	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1→𝐴)
89		f1imacnv 5656	. . . . . . . . . . . . . 14 ⊢ ((◡𝑓:𝐵–1-1→𝐴 ∧ 𝑧 ⊆ 𝐵) → (◡◡𝑓 “ (◡𝑓 “ 𝑧)) = 𝑧)
90	88, 89	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑧 ⊆ 𝐵) → (◡◡𝑓 “ (◡𝑓 “ 𝑧)) = 𝑧)
91	87, 90	eqtr3d 2273	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑧 ⊆ 𝐵) → (𝑓 “ (◡𝑓 “ 𝑧)) = 𝑧)
92	82, 91	sylan9eqr 2293	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑦 = (◡𝑓 “ 𝑧)) → (𝑓 “ 𝑦) = 𝑧)
93	92	eqcomd 2244	. . . . . . . . . 10 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑦 = (◡𝑓 “ 𝑧)) → 𝑧 = (𝑓 “ 𝑦))
94	93	ex 115	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑧 ⊆ 𝐵) → (𝑦 = (◡𝑓 “ 𝑧) → 𝑧 = (𝑓 “ 𝑦)))
95	81, 94	sylan2 286	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑧 ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶})) → (𝑦 = (◡𝑓 “ 𝑧) → 𝑧 = (𝑓 “ 𝑦)))
96	95	adantrl 482	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ∧ 𝑧 ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}))) → (𝑦 = (◡𝑓 “ 𝑧) → 𝑧 = (𝑓 “ 𝑦)))
97		simpl 109	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) → 𝑦 ∈ 𝒫 𝐴)
98	97	elpwid 3700	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) → 𝑦 ⊆ 𝐴)
99	32, 98	sylbi 121	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) → 𝑦 ⊆ 𝐴)
100		imaeq2 5122	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑓 “ 𝑦) → (◡𝑓 “ 𝑧) = (◡𝑓 “ (𝑓 “ 𝑦)))
101		f1imacnv 5656	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑦 ⊆ 𝐴) → (◡𝑓 “ (𝑓 “ 𝑦)) = 𝑦)
102	17, 101	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) → (◡𝑓 “ (𝑓 “ 𝑦)) = 𝑦)
103	100, 102	sylan9eqr 2293	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑧 = (𝑓 “ 𝑦)) → (◡𝑓 “ 𝑧) = 𝑦)
104	103	eqcomd 2244	. . . . . . . . . 10 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑧 = (𝑓 “ 𝑦)) → 𝑦 = (◡𝑓 “ 𝑧))
105	104	ex 115	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) → (𝑧 = (𝑓 “ 𝑦) → 𝑦 = (◡𝑓 “ 𝑧)))
106	99, 105	sylan2 286	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶})) → (𝑧 = (𝑓 “ 𝑦) → 𝑦 = (◡𝑓 “ 𝑧)))
107	106	adantrr 483	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ∧ 𝑧 ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}))) → (𝑧 = (𝑓 “ 𝑦) → 𝑦 = (◡𝑓 “ 𝑧)))
108	96, 107	impbid 129	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ∧ 𝑧 ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}))) → (𝑦 = (◡𝑓 “ 𝑧) ↔ 𝑧 = (𝑓 “ 𝑦)))
109	108	ex 115	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ((𝑦 ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ∧ 𝑧 ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶})) → (𝑦 = (◡𝑓 “ 𝑧) ↔ 𝑧 = (𝑓 “ 𝑦))))
110	8, 16, 45, 78, 109	en3d 7055	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ≈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}))
111	110	exlimiv 1651	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ≈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}))
112	1, 111	sylbi 121	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ≈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}))
113		df-pw 3690	. . . 4 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
114	113	ineq1i 3428	. . 3 ⊢ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) = ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∩ {𝑥 ∣ 𝑥 ≈ 𝐶})
115		inab 3499	. . 3 ⊢ ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐶)}
116	114, 115	eqtri 2259	. 2 ⊢ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐶)}
117		df-pw 3690	. . . 4 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵}
118	117	ineq1i 3428	. . 3 ⊢ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) = ({𝑥 ∣ 𝑥 ⊆ 𝐵} ∩ {𝑥 ∣ 𝑥 ≈ 𝐶})
119		inab 3499	. . 3 ⊢ ({𝑥 ∣ 𝑥 ⊆ 𝐵} ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) = {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝑥 ≈ 𝐶)}
120	118, 119	eqtri 2259	. 2 ⊢ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) = {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝑥 ≈ 𝐶)}
121	112, 116, 120	3brtr3g 4163	1 ⊢ (𝐴 ≈ 𝐵 → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐶)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝑥 ≈ 𝐶)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {cab 2224  Vcvv 2821   ∩ cin 3219   ⊆ wss 3220  𝒫 cpw 3688   class class class wbr 4130  ^◡ccnv 4773  dom cdm 4774  ran crn 4775   “ cima 4777  Rel wrel 4779  –1-1→wf1 5374  –onto→wfo 5375  –1-1-onto→wf1o 5376   ≈ cen 7020

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-er 6807  df-en 7023

This theorem is used by: (None)