MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1opw2 Structured version   Visualization version   GIF version

Theorem f1opw2 7661
Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 7662 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
f1opw2.1 (𝜑𝐹:𝐴1-1-onto𝐵)
f1opw2.2 (𝜑 → (𝐹𝑎) ∈ V)
f1opw2.3 (𝜑 → (𝐹𝑏) ∈ V)
Assertion
Ref Expression
f1opw2 (𝜑 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)):𝒫 𝐴1-1-onto→𝒫 𝐵)
Distinct variable groups:   𝑎,𝑏,𝐴   𝐵,𝑎,𝑏   𝐹,𝑎,𝑏   𝜑,𝑎,𝑏

Proof of Theorem f1opw2
StepHypRef Expression
1 eqid 2733 . 2 (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)) = (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏))
2 f1opw2.3 . . . 4 (𝜑 → (𝐹𝑏) ∈ V)
3 imassrn 6071 . . . . 5 (𝐹𝑏) ⊆ ran 𝐹
4 f1opw2.1 . . . . . . 7 (𝜑𝐹:𝐴1-1-onto𝐵)
5 f1ofo 6841 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
64, 5syl 17 . . . . . 6 (𝜑𝐹:𝐴onto𝐵)
7 forn 6809 . . . . . 6 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
86, 7syl 17 . . . . 5 (𝜑 → ran 𝐹 = 𝐵)
93, 8sseqtrid 4035 . . . 4 (𝜑 → (𝐹𝑏) ⊆ 𝐵)
102, 9elpwd 4609 . . 3 (𝜑 → (𝐹𝑏) ∈ 𝒫 𝐵)
1110adantr 482 . 2 ((𝜑𝑏 ∈ 𝒫 𝐴) → (𝐹𝑏) ∈ 𝒫 𝐵)
12 f1opw2.2 . . . 4 (𝜑 → (𝐹𝑎) ∈ V)
13 imassrn 6071 . . . . 5 (𝐹𝑎) ⊆ ran 𝐹
14 dfdm4 5896 . . . . . 6 dom 𝐹 = ran 𝐹
15 f1odm 6838 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵 → dom 𝐹 = 𝐴)
164, 15syl 17 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
1714, 16eqtr3id 2787 . . . . 5 (𝜑 → ran 𝐹 = 𝐴)
1813, 17sseqtrid 4035 . . . 4 (𝜑 → (𝐹𝑎) ⊆ 𝐴)
1912, 18elpwd 4609 . . 3 (𝜑 → (𝐹𝑎) ∈ 𝒫 𝐴)
2019adantr 482 . 2 ((𝜑𝑎 ∈ 𝒫 𝐵) → (𝐹𝑎) ∈ 𝒫 𝐴)
21 elpwi 4610 . . . . . . 7 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
2221adantl 483 . . . . . 6 ((𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵) → 𝑎𝐵)
23 foimacnv 6851 . . . . . 6 ((𝐹:𝐴onto𝐵𝑎𝐵) → (𝐹 “ (𝐹𝑎)) = 𝑎)
246, 22, 23syl2an 597 . . . . 5 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝐹 “ (𝐹𝑎)) = 𝑎)
2524eqcomd 2739 . . . 4 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → 𝑎 = (𝐹 “ (𝐹𝑎)))
26 imaeq2 6056 . . . . 5 (𝑏 = (𝐹𝑎) → (𝐹𝑏) = (𝐹 “ (𝐹𝑎)))
2726eqeq2d 2744 . . . 4 (𝑏 = (𝐹𝑎) → (𝑎 = (𝐹𝑏) ↔ 𝑎 = (𝐹 “ (𝐹𝑎))))
2825, 27syl5ibrcom 246 . . 3 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (𝐹𝑎) → 𝑎 = (𝐹𝑏)))
29 f1of1 6833 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
304, 29syl 17 . . . . . 6 (𝜑𝐹:𝐴1-1𝐵)
31 elpwi 4610 . . . . . . 7 (𝑏 ∈ 𝒫 𝐴𝑏𝐴)
3231adantr 482 . . . . . 6 ((𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵) → 𝑏𝐴)
33 f1imacnv 6850 . . . . . 6 ((𝐹:𝐴1-1𝐵𝑏𝐴) → (𝐹 “ (𝐹𝑏)) = 𝑏)
3430, 32, 33syl2an 597 . . . . 5 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝐹 “ (𝐹𝑏)) = 𝑏)
3534eqcomd 2739 . . . 4 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → 𝑏 = (𝐹 “ (𝐹𝑏)))
36 imaeq2 6056 . . . . 5 (𝑎 = (𝐹𝑏) → (𝐹𝑎) = (𝐹 “ (𝐹𝑏)))
3736eqeq2d 2744 . . . 4 (𝑎 = (𝐹𝑏) → (𝑏 = (𝐹𝑎) ↔ 𝑏 = (𝐹 “ (𝐹𝑏))))
3835, 37syl5ibrcom 246 . . 3 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝑎 = (𝐹𝑏) → 𝑏 = (𝐹𝑎)))
3928, 38impbid 211 . 2 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (𝐹𝑎) ↔ 𝑎 = (𝐹𝑏)))
401, 11, 20, 39f1o2d 7660 1 (𝜑 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)):𝒫 𝐴1-1-onto→𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  wss 3949  𝒫 cpw 4603  cmpt 5232  ccnv 5676  dom cdm 5677  ran crn 5678  cima 5680  1-1wf1 6541  ontowfo 6542  1-1-ontowf1o 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551
This theorem is referenced by:  f1opw  7662
  Copyright terms: Public domain W3C validator