Theorem f1opw2 6044

Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6045 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
f1opw2.1	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
f1opw2.2	⊢ (𝜑 → (◡𝐹 “ 𝑎) ∈ V)
f1opw2.3	⊢ (𝜑 → (𝐹 “ 𝑏) ∈ V)

Assertion

Ref	Expression
f1opw2	⊢ (𝜑 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵)

Distinct variable groups:   𝑎,𝑏,𝐴   𝐵,𝑎,𝑏   𝐹,𝑎,𝑏   𝜑,𝑎,𝑏

Proof of Theorem f1opw2

Step	Hyp	Ref	Expression
1		eqid 2165	. 2 ⊢ (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)) = (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏))
2		imassrn 4957	. . . . 5 ⊢ (𝐹 “ 𝑏) ⊆ ran 𝐹
3		f1opw2.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
4		f1ofo 5439	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)
6		forn 5413	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
7	5, 6	syl 14	. . . . 5 ⊢ (𝜑 → ran 𝐹 = 𝐵)
8	2, 7	sseqtrid 3192	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝑏) ⊆ 𝐵)
9		f1opw2.3	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝑏) ∈ V)
10		elpwg 3567	. . . . 5 ⊢ ((𝐹 “ 𝑏) ∈ V → ((𝐹 “ 𝑏) ∈ 𝒫 𝐵 ↔ (𝐹 “ 𝑏) ⊆ 𝐵))
11	9, 10	syl 14	. . . 4 ⊢ (𝜑 → ((𝐹 “ 𝑏) ∈ 𝒫 𝐵 ↔ (𝐹 “ 𝑏) ⊆ 𝐵))
12	8, 11	mpbird 166	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑏) ∈ 𝒫 𝐵)
13	12	adantr 274	. 2 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝒫 𝐴) → (𝐹 “ 𝑏) ∈ 𝒫 𝐵)
14		imassrn 4957	. . . . 5 ⊢ (◡𝐹 “ 𝑎) ⊆ ran ◡𝐹
15		dfdm4 4796	. . . . . 6 ⊢ dom 𝐹 = ran ◡𝐹
16		f1odm 5436	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴)
17	3, 16	syl 14	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
18	15, 17	eqtr3id 2213	. . . . 5 ⊢ (𝜑 → ran ◡𝐹 = 𝐴)
19	14, 18	sseqtrid 3192	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑎) ⊆ 𝐴)
20		f1opw2.2	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ 𝑎) ∈ V)
21		elpwg 3567	. . . . 5 ⊢ ((◡𝐹 “ 𝑎) ∈ V → ((◡𝐹 “ 𝑎) ∈ 𝒫 𝐴 ↔ (◡𝐹 “ 𝑎) ⊆ 𝐴))
22	20, 21	syl 14	. . . 4 ⊢ (𝜑 → ((◡𝐹 “ 𝑎) ∈ 𝒫 𝐴 ↔ (◡𝐹 “ 𝑎) ⊆ 𝐴))
23	19, 22	mpbird 166	. . 3 ⊢ (𝜑 → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴)
24	23	adantr 274	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝐵) → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴)
25		elpwi 3568	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝐵 → 𝑎 ⊆ 𝐵)
26	25	adantl 275	. . . . . 6 ⊢ ((𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵) → 𝑎 ⊆ 𝐵)
27		foimacnv 5450	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑎 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
28	5, 26, 27	syl2an 287	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
29	28	eqcomd 2171	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → 𝑎 = (𝐹 “ (◡𝐹 “ 𝑎)))
30		imaeq2 4942	. . . . 5 ⊢ (𝑏 = (◡𝐹 “ 𝑎) → (𝐹 “ 𝑏) = (𝐹 “ (◡𝐹 “ 𝑎)))
31	30	eqeq2d 2177	. . . 4 ⊢ (𝑏 = (◡𝐹 “ 𝑎) → (𝑎 = (𝐹 “ 𝑏) ↔ 𝑎 = (𝐹 “ (◡𝐹 “ 𝑎))))
32	29, 31	syl5ibrcom 156	. . 3 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (◡𝐹 “ 𝑎) → 𝑎 = (𝐹 “ 𝑏)))
33		f1of1 5431	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)
34	3, 33	syl 14	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
35		elpwi 3568	. . . . . . 7 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
36	35	adantr 274	. . . . . 6 ⊢ ((𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵) → 𝑏 ⊆ 𝐴)
37		f1imacnv 5449	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑏 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝑏)) = 𝑏)
38	34, 36, 37	syl2an 287	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (◡𝐹 “ (𝐹 “ 𝑏)) = 𝑏)
39	38	eqcomd 2171	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → 𝑏 = (◡𝐹 “ (𝐹 “ 𝑏)))
40		imaeq2 4942	. . . . 5 ⊢ (𝑎 = (𝐹 “ 𝑏) → (◡𝐹 “ 𝑎) = (◡𝐹 “ (𝐹 “ 𝑏)))
41	40	eqeq2d 2177	. . . 4 ⊢ (𝑎 = (𝐹 “ 𝑏) → (𝑏 = (◡𝐹 “ 𝑎) ↔ 𝑏 = (◡𝐹 “ (𝐹 “ 𝑏))))
42	39, 41	syl5ibrcom 156	. . 3 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝑎 = (𝐹 “ 𝑏) → 𝑏 = (◡𝐹 “ 𝑎)))
43	32, 42	impbid 128	. 2 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (◡𝐹 “ 𝑎) ↔ 𝑎 = (𝐹 “ 𝑏)))
44	1, 13, 24, 43	f1o2d 6043	1 ⊢ (𝜑 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116  𝒫 cpw 3559   ↦ cmpt 4043  ^◡ccnv 4603  dom cdm 4604  ran crn 4605   “ cima 4607  –1-1→wf1 5185  –onto→wfo 5186  –1-1-onto→wf1o 5187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195

This theorem is referenced by:  f1opw  6045