Theorem f1opw2 7644

Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 7645 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 2729	. 2 ⊢ (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)) = (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏))
2		f1opw2.3	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝑏) ∈ V)
3		imassrn 6042	. . . . 5 ⊢ (𝐹 “ 𝑏) ⊆ ran 𝐹
4		f1opw2.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
5		f1ofo 6807	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
6	4, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)
7		forn 6775	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐹 = 𝐵)
9	3, 8	sseqtrid 3989	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝑏) ⊆ 𝐵)
10	2, 9	elpwd 4569	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑏) ∈ 𝒫 𝐵)
11	10	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝒫 𝐴) → (𝐹 “ 𝑏) ∈ 𝒫 𝐵)
12		f1opw2.2	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑎) ∈ V)
13		imassrn 6042	. . . . 5 ⊢ (◡𝐹 “ 𝑎) ⊆ ran ◡𝐹
14		dfdm4 5859	. . . . . 6 ⊢ dom 𝐹 = ran ◡𝐹
15		f1odm 6804	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴)
16	4, 15	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
17	14, 16	eqtr3id 2778	. . . . 5 ⊢ (𝜑 → ran ◡𝐹 = 𝐴)
18	13, 17	sseqtrid 3989	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑎) ⊆ 𝐴)
19	12, 18	elpwd 4569	. . 3 ⊢ (𝜑 → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴)
20	19	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝐵) → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴)
21		elpwi 4570	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝐵 → 𝑎 ⊆ 𝐵)
22	21	adantl 481	. . . . . 6 ⊢ ((𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵) → 𝑎 ⊆ 𝐵)
23		foimacnv 6817	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑎 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
24	6, 22, 23	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
25	24	eqcomd 2735	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → 𝑎 = (𝐹 “ (◡𝐹 “ 𝑎)))
26		imaeq2 6027	. . . . 5 ⊢ (𝑏 = (◡𝐹 “ 𝑎) → (𝐹 “ 𝑏) = (𝐹 “ (◡𝐹 “ 𝑎)))
27	26	eqeq2d 2740	. . . 4 ⊢ (𝑏 = (◡𝐹 “ 𝑎) → (𝑎 = (𝐹 “ 𝑏) ↔ 𝑎 = (𝐹 “ (◡𝐹 “ 𝑎))))
28	25, 27	syl5ibrcom 247	. . 3 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (◡𝐹 “ 𝑎) → 𝑎 = (𝐹 “ 𝑏)))
29		f1of1 6799	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)
30	4, 29	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
31		elpwi 4570	. . . . . . 7 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
32	31	adantr 480	. . . . . 6 ⊢ ((𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵) → 𝑏 ⊆ 𝐴)
33		f1imacnv 6816	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑏 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝑏)) = 𝑏)
34	30, 32, 33	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (◡𝐹 “ (𝐹 “ 𝑏)) = 𝑏)
35	34	eqcomd 2735	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → 𝑏 = (◡𝐹 “ (𝐹 “ 𝑏)))
36		imaeq2 6027	. . . . 5 ⊢ (𝑎 = (𝐹 “ 𝑏) → (◡𝐹 “ 𝑎) = (◡𝐹 “ (𝐹 “ 𝑏)))
37	36	eqeq2d 2740	. . . 4 ⊢ (𝑎 = (𝐹 “ 𝑏) → (𝑏 = (◡𝐹 “ 𝑎) ↔ 𝑏 = (◡𝐹 “ (𝐹 “ 𝑏))))
38	35, 37	syl5ibrcom 247	. . 3 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝑎 = (𝐹 “ 𝑏) → 𝑏 = (◡𝐹 “ 𝑎)))
39	28, 38	impbid 212	. 2 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (◡𝐹 “ 𝑎) ↔ 𝑎 = (𝐹 “ 𝑏)))
40	1, 11, 20, 39	f1o2d 7643	1 ⊢ (𝜑 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ⊆ wss 3914  𝒫 cpw 4563   ↦ cmpt 5188  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   “ cima 5641  –1-1→wf1 6508  –onto→wfo 6509  –1-1-onto→wf1o 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518

This theorem is referenced by:  f1opw  7645