Theorem f1opw2 7607

Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 7608 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 2733	. 2 ⊢ (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)) = (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏))
2		f1opw2.3	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝑏) ∈ V)
3		imassrn 6024	. . . . 5 ⊢ (𝐹 “ 𝑏) ⊆ ran 𝐹
4		f1opw2.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
5		f1ofo 6775	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
6	4, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)
7		forn 6743	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐹 = 𝐵)
9	3, 8	sseqtrid 3973	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝑏) ⊆ 𝐵)
10	2, 9	elpwd 4555	. . 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 6024	. . . . 5 ⊢ (◡𝐹 “ 𝑎) ⊆ ran ◡𝐹
14		dfdm4 5839	. . . . . 6 ⊢ dom 𝐹 = ran ◡𝐹
15		f1odm 6772	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴)
16	4, 15	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
17	14, 16	eqtr3id 2782	. . . . 5 ⊢ (𝜑 → ran ◡𝐹 = 𝐴)
18	13, 17	sseqtrid 3973	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑎) ⊆ 𝐴)
19	12, 18	elpwd 4555	. . 3 ⊢ (𝜑 → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴)
20	19	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝐵) → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴)
21		elpwi 4556	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝐵 → 𝑎 ⊆ 𝐵)
22	21	adantl 481	. . . . . 6 ⊢ ((𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵) → 𝑎 ⊆ 𝐵)
23		foimacnv 6785	. . . . . 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 2739	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → 𝑎 = (𝐹 “ (◡𝐹 “ 𝑎)))
26		imaeq2 6009	. . . . 5 ⊢ (𝑏 = (◡𝐹 “ 𝑎) → (𝐹 “ 𝑏) = (𝐹 “ (◡𝐹 “ 𝑎)))
27	26	eqeq2d 2744	. . . 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 6767	. . . . . . 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 4556	. . . . . . 7 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
32	31	adantr 480	. . . . . 6 ⊢ ((𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵) → 𝑏 ⊆ 𝐴)
33		f1imacnv 6784	. . . . . 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 2739	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → 𝑏 = (◡𝐹 “ (𝐹 “ 𝑏)))
36		imaeq2 6009	. . . . 5 ⊢ (𝑎 = (𝐹 “ 𝑏) → (◡𝐹 “ 𝑎) = (◡𝐹 “ (𝐹 “ 𝑏)))
37	36	eqeq2d 2744	. . . 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 7606	1 ⊢ (𝜑 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ⊆ wss 3898  𝒫 cpw 4549   ↦ cmpt 5174  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   “ cima 5622  –1-1→wf1 6483  –onto→wfo 6484  –1-1-onto→wf1o 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493

This theorem is referenced by:  f1opw  7608