Theorem f1opw2 5726

Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 5727 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 2081	. 2 ⊢ (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)) = (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏))
2		imassrn 4699	. . . . 5 ⊢ (𝐹 “ 𝑏) ⊆ ran 𝐹
3		f1opw2.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
4		f1ofo 5153	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)
6		forn 5129	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
7	5, 6	syl 14	. . . . 5 ⊢ (𝜑 → ran 𝐹 = 𝐵)
8	2, 7	syl5sseq 3047	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝑏) ⊆ 𝐵)
9		f1opw2.3	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝑏) ∈ V)
10		elpwg 3390	. . . . 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 165	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑏) ∈ 𝒫 𝐵)
13	12	adantr 270	. 2 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝒫 𝐴) → (𝐹 “ 𝑏) ∈ 𝒫 𝐵)
14		imassrn 4699	. . . . 5 ⊢ (◡𝐹 “ 𝑎) ⊆ ran ◡𝐹
15		dfdm4 4545	. . . . . 6 ⊢ dom 𝐹 = ran ◡𝐹
16		f1odm 5150	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴)
17	3, 16	syl 14	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
18	15, 17	syl5eqr 2127	. . . . 5 ⊢ (𝜑 → ran ◡𝐹 = 𝐴)
19	14, 18	syl5sseq 3047	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑎) ⊆ 𝐴)
20		f1opw2.2	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ 𝑎) ∈ V)
21		elpwg 3390	. . . . 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 165	. . 3 ⊢ (𝜑 → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴)
24	23	adantr 270	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝐵) → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴)
25		elpwi 3391	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝐵 → 𝑎 ⊆ 𝐵)
26	25	adantl 271	. . . . . 6 ⊢ ((𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵) → 𝑎 ⊆ 𝐵)
27		foimacnv 5164	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑎 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
28	5, 26, 27	syl2an 283	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
29	28	eqcomd 2086	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → 𝑎 = (𝐹 “ (◡𝐹 “ 𝑎)))
30		imaeq2 4684	. . . . 5 ⊢ (𝑏 = (◡𝐹 “ 𝑎) → (𝐹 “ 𝑏) = (𝐹 “ (◡𝐹 “ 𝑎)))
31	30	eqeq2d 2092	. . . 4 ⊢ (𝑏 = (◡𝐹 “ 𝑎) → (𝑎 = (𝐹 “ 𝑏) ↔ 𝑎 = (𝐹 “ (◡𝐹 “ 𝑎))))
32	29, 31	syl5ibrcom 155	. . 3 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (◡𝐹 “ 𝑎) → 𝑎 = (𝐹 “ 𝑏)))
33		f1of1 5145	. . . . . . 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 3391	. . . . . . 7 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
36	35	adantr 270	. . . . . 6 ⊢ ((𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵) → 𝑏 ⊆ 𝐴)
37		f1imacnv 5163	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑏 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝑏)) = 𝑏)
38	34, 36, 37	syl2an 283	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (◡𝐹 “ (𝐹 “ 𝑏)) = 𝑏)
39	38	eqcomd 2086	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → 𝑏 = (◡𝐹 “ (𝐹 “ 𝑏)))
40		imaeq2 4684	. . . . 5 ⊢ (𝑎 = (𝐹 “ 𝑏) → (◡𝐹 “ 𝑎) = (◡𝐹 “ (𝐹 “ 𝑏)))
41	40	eqeq2d 2092	. . . 4 ⊢ (𝑎 = (𝐹 “ 𝑏) → (𝑏 = (◡𝐹 “ 𝑎) ↔ 𝑏 = (◡𝐹 “ (𝐹 “ 𝑏))))
42	39, 41	syl5ibrcom 155	. . 3 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝑎 = (𝐹 “ 𝑏) → 𝑏 = (◡𝐹 “ 𝑎)))
43	32, 42	impbid 127	. 2 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (◡𝐹 “ 𝑎) ↔ 𝑎 = (𝐹 “ 𝑏)))
44	1, 13, 24, 43	f1o2d 5725	1 ⊢ (𝜑 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ⊆ wss 2973  𝒫 cpw 3382   ↦ cmpt 3839  ^◡ccnv 4362  dom cdm 4363  ran crn 4364   “ cima 4366  –1-1→wf1 4919  –onto→wfo 4920  –1-1-onto→wf1o 4921

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929

This theorem is referenced by:  f1opw  5727