Theorem ssenen 9170

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 8983	. . 3 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2		f1odm 6828	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → dom 𝑓 = 𝐴)
3		vex 3455	. . . . . . . 8 ⊢ 𝑓 ∈ V
4	3	dmex 7921	. . . . . . 7 ⊢ dom 𝑓 ∈ V
5	2, 4	eqeltrrdi 2870	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐴 ∈ V)
6		pwexg 5340	. . . . . 6 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
7		inex1g 5279	. . . . . 6 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ∈ V)
8	5, 6, 7	3syl 19	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ∈ V)
9		f1ofo 6832	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
10		forn 6799	. . . . . . . 8 ⊢ (𝑓:𝐴–onto→𝐵 → ran 𝑓 = 𝐵)
11	9, 10	syl 18	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ran 𝑓 = 𝐵)
12	3	rnex 7922	. . . . . . 7 ⊢ ran 𝑓 ∈ V
13	11, 12	eqeltrrdi 2870	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐵 ∈ V)
14		pwexg 5340	. . . . . 6 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
15		inex1g 5279	. . . . . 6 ⊢ (𝒫 𝐵 ∈ V → (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ∈ V)
16	13, 14, 15	3syl 19	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ∈ V)
17		f1of1 6823	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–1-1→𝐵)
18	17	adantr 486	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) → 𝑓:𝐴–1-1→𝐵)
19	13	adantr 486	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) → 𝐵 ∈ V)
20		simpr 490	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
21		vex 3455	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	21	a1i 11	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ V)
23		f1imaen2g 9042	. . . . . . . . . 10 ⊢ (((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ∈ V) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑦 ∈ V)) → (𝑓 “ 𝑦) ≈ 𝑦)
24	18, 19, 20, 22, 23	syl22anc 852	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) → (𝑓 “ 𝑦) ≈ 𝑦)
25		entr 9033	. . . . . . . . 9 ⊢ (((𝑓 “ 𝑦) ≈ 𝑦 ∧ 𝑦 ≈ 𝐶) → (𝑓 “ 𝑦) ≈ 𝐶)
26	24, 25	sylan 592	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑦 ≈ 𝐶) → (𝑓 “ 𝑦) ≈ 𝐶)
27	26	expl 463	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ((𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝐶) → (𝑓 “ 𝑦) ≈ 𝐶))
28		imassrn 6197	. . . . . . . . 9 ⊢ (𝑓 “ 𝑦) ⊆ ran 𝑓
29	28, 10	sseqtrid 3973	. . . . . . . 8 ⊢ (𝑓:𝐴–onto→𝐵 → (𝑓 “ 𝑦) ⊆ 𝐵)
30	9, 29	syl 18	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑓 “ 𝑦) ⊆ 𝐵)
31	27, 30	jctild 535	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ((𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝐶) → ((𝑓 “ 𝑦) ⊆ 𝐵 ∧ (𝑓 “ 𝑦) ≈ 𝐶)))
32		elin 3915	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}))
33	21	elpw 4561	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
34		breq1 5106	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝐶 ↔ 𝑦 ≈ 𝐶))
35	21, 34	elab 3633	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶} ↔ 𝑦 ≈ 𝐶)
36	33, 35	anbi12i 640	. . . . . . 7 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝐶))
37	32, 36	bitri 278	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝐶))
38		elin 3915	. . . . . . 7 ⊢ ((𝑓 “ 𝑦) ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ ((𝑓 “ 𝑦) ∈ 𝒫 𝐵 ∧ (𝑓 “ 𝑦) ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}))
39	3	imaex 7926	. . . . . . . . 9 ⊢ (𝑓 “ 𝑦) ∈ V
40	39	elpw 4561	. . . . . . . 8 ⊢ ((𝑓 “ 𝑦) ∈ 𝒫 𝐵 ↔ (𝑓 “ 𝑦) ⊆ 𝐵)
41		breq1 5106	. . . . . . . . 9 ⊢ (𝑥 = (𝑓 “ 𝑦) → (𝑥 ≈ 𝐶 ↔ (𝑓 “ 𝑦) ≈ 𝐶))
42	39, 41	elab 3633	. . . . . . . 8 ⊢ ((𝑓 “ 𝑦) ∈ {𝑥 ∣ 𝑥 ≈ 𝐶} ↔ (𝑓 “ 𝑦) ≈ 𝐶)
43	40, 42	anbi12i 640	. . . . . . 7 ⊢ (((𝑓 “ 𝑦) ∈ 𝒫 𝐵 ∧ (𝑓 “ 𝑦) ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ ((𝑓 “ 𝑦) ⊆ 𝐵 ∧ (𝑓 “ 𝑦) ≈ 𝐶))
44	38, 43	bitri 278	. . . . . 6 ⊢ ((𝑓 “ 𝑦) ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ ((𝑓 “ 𝑦) ⊆ 𝐵 ∧ (𝑓 “ 𝑦) ≈ 𝐶))
45	31, 37, 44	3imtr4g 299	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) → (𝑓 “ 𝑦) ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶})))
46		f1ocnv 6837	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1-onto→𝐴)
47		f1of1 6823	. . . . . . . . . . . 12 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → ◡𝑓:𝐵–1-1→𝐴)
48		f1f1orn 6836	. . . . . . . . . . . 12 ⊢ (◡𝑓:𝐵–1-1→𝐴 → ◡𝑓:𝐵–1-1-onto→ran ◡𝑓)
49		f1of1 6823	. . . . . . . . . . . 12 ⊢ (◡𝑓:𝐵–1-1-onto→ran ◡𝑓 → ◡𝑓:𝐵–1-1→ran ◡𝑓)
50	47, 48, 49	3syl 19	. . . . . . . . . . 11 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → ◡𝑓:𝐵–1-1→ran ◡𝑓)
51		vex 3455	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
52	51	f1imaen 9044	. . . . . . . . . . 11 ⊢ ((◡𝑓:𝐵–1-1→ran ◡𝑓 ∧ 𝑧 ⊆ 𝐵) → (◡𝑓 “ 𝑧) ≈ 𝑧)
53	50, 52	sylan 592	. . . . . . . . . 10 ⊢ ((◡𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑧 ⊆ 𝐵) → (◡𝑓 “ 𝑧) ≈ 𝑧)
54		entr 9033	. . . . . . . . . 10 ⊢ (((◡𝑓 “ 𝑧) ≈ 𝑧 ∧ 𝑧 ≈ 𝐶) → (◡𝑓 “ 𝑧) ≈ 𝐶)
55	53, 54	sylan 592	. . . . . . . . 9 ⊢ (((◡𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑧 ≈ 𝐶) → (◡𝑓 “ 𝑧) ≈ 𝐶)
56	55	expl 463	. . . . . . . 8 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → ((𝑧 ⊆ 𝐵 ∧ 𝑧 ≈ 𝐶) → (◡𝑓 “ 𝑧) ≈ 𝐶))
57		f1ofo 6832	. . . . . . . . 9 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → ◡𝑓:𝐵–onto→𝐴)
58		imassrn 6197	. . . . . . . . . 10 ⊢ (◡𝑓 “ 𝑧) ⊆ ran ◡𝑓
59		forn 6799	. . . . . . . . . 10 ⊢ (◡𝑓:𝐵–onto→𝐴 → ran ◡𝑓 = 𝐴)
60	58, 59	sseqtrid 3973	. . . . . . . . 9 ⊢ (◡𝑓:𝐵–onto→𝐴 → (◡𝑓 “ 𝑧) ⊆ 𝐴)
61	57, 60	syl 18	. . . . . . . 8 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → (◡𝑓 “ 𝑧) ⊆ 𝐴)
62	56, 61	jctild 535	. . . . . . 7 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → ((𝑧 ⊆ 𝐵 ∧ 𝑧 ≈ 𝐶) → ((◡𝑓 “ 𝑧) ⊆ 𝐴 ∧ (◡𝑓 “ 𝑧) ≈ 𝐶)))
63	46, 62	syl 18	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ((𝑧 ⊆ 𝐵 ∧ 𝑧 ≈ 𝐶) → ((◡𝑓 “ 𝑧) ⊆ 𝐴 ∧ (◡𝑓 “ 𝑧) ≈ 𝐶)))
64		elin 3915	. . . . . . 7 ⊢ (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ (𝑧 ∈ 𝒫 𝐵 ∧ 𝑧 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}))
65	51	elpw 4561	. . . . . . . 8 ⊢ (𝑧 ∈ 𝒫 𝐵 ↔ 𝑧 ⊆ 𝐵)
66		breq1 5106	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ≈ 𝐶 ↔ 𝑧 ≈ 𝐶))
67	51, 66	elab 3633	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶} ↔ 𝑧 ≈ 𝐶)
68	65, 67	anbi12i 640	. . . . . . 7 ⊢ ((𝑧 ∈ 𝒫 𝐵 ∧ 𝑧 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ (𝑧 ⊆ 𝐵 ∧ 𝑧 ≈ 𝐶))
69	64, 68	bitri 278	. . . . . 6 ⊢ (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ (𝑧 ⊆ 𝐵 ∧ 𝑧 ≈ 𝐶))
70		elin 3915	. . . . . . 7 ⊢ ((◡𝑓 “ 𝑧) ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ ((◡𝑓 “ 𝑧) ∈ 𝒫 𝐴 ∧ (◡𝑓 “ 𝑧) ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}))
71	3	cnvex 7937	. . . . . . . . . 10 ⊢ ◡𝑓 ∈ V
72	71	imaex 7926	. . . . . . . . 9 ⊢ (◡𝑓 “ 𝑧) ∈ V
73	72	elpw 4561	. . . . . . . 8 ⊢ ((◡𝑓 “ 𝑧) ∈ 𝒫 𝐴 ↔ (◡𝑓 “ 𝑧) ⊆ 𝐴)
74		breq1 5106	. . . . . . . . 9 ⊢ (𝑥 = (◡𝑓 “ 𝑧) → (𝑥 ≈ 𝐶 ↔ (◡𝑓 “ 𝑧) ≈ 𝐶))
75	72, 74	elab 3633	. . . . . . . 8 ⊢ ((◡𝑓 “ 𝑧) ∈ {𝑥 ∣ 𝑥 ≈ 𝐶} ↔ (◡𝑓 “ 𝑧) ≈ 𝐶)
76	73, 75	anbi12i 640	. . . . . . 7 ⊢ (((◡𝑓 “ 𝑧) ∈ 𝒫 𝐴 ∧ (◡𝑓 “ 𝑧) ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ ((◡𝑓 “ 𝑧) ⊆ 𝐴 ∧ (◡𝑓 “ 𝑧) ≈ 𝐶))
77	70, 76	bitri 278	. . . . . 6 ⊢ ((◡𝑓 “ 𝑧) ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ↔ ((◡𝑓 “ 𝑧) ⊆ 𝐴 ∧ (◡𝑓 “ 𝑧) ≈ 𝐶))
78	63, 69, 77	3imtr4g 299	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) → (◡𝑓 “ 𝑧) ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶})))
79		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝒫 𝐵 ∧ 𝑧 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) → 𝑧 ∈ 𝒫 𝐵)
80	79	elpwid 4566	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝒫 𝐵 ∧ 𝑧 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) → 𝑧 ⊆ 𝐵)
81	64, 80	sylbi 220	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) → 𝑧 ⊆ 𝐵)
82		imaeq2 6048	. . . . . . . . . . . 12 ⊢ (𝑦 = (◡𝑓 “ 𝑧) → (𝑓 “ 𝑦) = (𝑓 “ (◡𝑓 “ 𝑧)))
83		f1orel 6827	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → Rel 𝑓)
84		dfrel2 6181	. . . . . . . . . . . . . . . 16 ⊢ (Rel 𝑓 ↔ ◡◡𝑓 = 𝑓)
85	83, 84	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡◡𝑓 = 𝑓)
86	85	imaeq1d 6051	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (◡◡𝑓 “ (◡𝑓 “ 𝑧)) = (𝑓 “ (◡𝑓 “ 𝑧)))
87	86	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑧 ⊆ 𝐵) → (◡◡𝑓 “ (◡𝑓 “ 𝑧)) = (𝑓 “ (◡𝑓 “ 𝑧)))
88	46, 47	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1→𝐴)
89		f1imacnv 6841	. . . . . . . . . . . . . 14 ⊢ ((◡𝑓:𝐵–1-1→𝐴 ∧ 𝑧 ⊆ 𝐵) → (◡◡𝑓 “ (◡𝑓 “ 𝑧)) = 𝑧)
90	88, 89	sylan 592	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑧 ⊆ 𝐵) → (◡◡𝑓 “ (◡𝑓 “ 𝑧)) = 𝑧)
91	87, 90	eqtr3d 2798	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑧 ⊆ 𝐵) → (𝑓 “ (◡𝑓 “ 𝑧)) = 𝑧)
92	82, 91	sylan9eqr 2818	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑦 = (◡𝑓 “ 𝑧)) → (𝑓 “ 𝑦) = 𝑧)
93	92	eqcomd 2767	. . . . . . . . . 10 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑦 = (◡𝑓 “ 𝑧)) → 𝑧 = (𝑓 “ 𝑦))
94	93	ex 418	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑧 ⊆ 𝐵) → (𝑦 = (◡𝑓 “ 𝑧) → 𝑧 = (𝑓 “ 𝑦)))
95	81, 94	sylan2 605	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑧 ∈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶})) → (𝑦 = (◡𝑓 “ 𝑧) → 𝑧 = (𝑓 “ 𝑦)))
96	95	adantrl 729	. . . . . . 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 488	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) → 𝑦 ∈ 𝒫 𝐴)
98	97	elpwid 4566	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝑥 ≈ 𝐶}) → 𝑦 ⊆ 𝐴)
99	32, 98	sylbi 220	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) → 𝑦 ⊆ 𝐴)
100		imaeq2 6048	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑓 “ 𝑦) → (◡𝑓 “ 𝑧) = (◡𝑓 “ (𝑓 “ 𝑦)))
101		f1imacnv 6841	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑦 ⊆ 𝐴) → (◡𝑓 “ (𝑓 “ 𝑦)) = 𝑦)
102	17, 101	sylan 592	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) → (◡𝑓 “ (𝑓 “ 𝑦)) = 𝑦)
103	100, 102	sylan9eqr 2818	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑧 = (𝑓 “ 𝑦)) → (◡𝑓 “ 𝑧) = 𝑦)
104	103	eqcomd 2767	. . . . . . . . . 10 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑧 = (𝑓 “ 𝑦)) → 𝑦 = (◡𝑓 “ 𝑧))
105	104	ex 418	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ⊆ 𝐴) → (𝑧 = (𝑓 “ 𝑦) → 𝑦 = (◡𝑓 “ 𝑧)))
106	99, 105	sylan2 605	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶})) → (𝑧 = (𝑓 “ 𝑦) → 𝑦 = (◡𝑓 “ 𝑧)))
107	106	adantrr 730	. . . . . . 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 215	. . . . . 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 418	. . . . 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 9016	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ≈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}))
111	110	exlimiv 1963	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ≈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}))
112	1, 111	sylbi 220	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) ≈ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}))
113		df-pw 4559	. . . 4 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
114	113	ineq1i 4162	. . 3 ⊢ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) = ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∩ {𝑥 ∣ 𝑥 ≈ 𝐶})
115		inab 4255	. . 3 ⊢ ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐶)}
116	114, 115	eqtri 2784	. 2 ⊢ (𝒫 𝐴 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐶)}
117		df-pw 4559	. . . 4 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵}
118	117	ineq1i 4162	. . 3 ⊢ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) = ({𝑥 ∣ 𝑥 ⊆ 𝐵} ∩ {𝑥 ∣ 𝑥 ≈ 𝐶})
119		inab 4255	. . 3 ⊢ ({𝑥 ∣ 𝑥 ⊆ 𝐵} ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) = {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝑥 ≈ 𝐶)}
120	118, 119	eqtri 2784	. 2 ⊢ (𝒫 𝐵 ∩ {𝑥 ∣ 𝑥 ≈ 𝐶}) = {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝑥 ≈ 𝐶)}
121	112, 116, 120	3brtr3g 5138	1 ⊢ (𝐴 ≈ 𝐵 → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐶)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝑥 ≈ 𝐶)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557   class class class wbr 5103  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   “ cima 5654  Rel wrel 5656  –1-1→wf1 6535  –onto→wfo 6536  –1-1-onto→wf1o 6537   ≈ cen 8970

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-er 8717  df-en 8974

This theorem is used by:  infmap2  10295