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Theorem f1opw2 7566
Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 7567 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 2737 . 2 (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)) = (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏))
2 f1opw2.3 . . . 4 (𝜑 → (𝐹𝑏) ∈ V)
3 imassrn 5998 . . . . 5 (𝐹𝑏) ⊆ ran 𝐹
4 f1opw2.1 . . . . . . 7 (𝜑𝐹:𝐴1-1-onto𝐵)
5 f1ofo 6761 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
64, 5syl 17 . . . . . 6 (𝜑𝐹:𝐴onto𝐵)
7 forn 6729 . . . . . 6 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
86, 7syl 17 . . . . 5 (𝜑 → ran 𝐹 = 𝐵)
93, 8sseqtrid 3983 . . . 4 (𝜑 → (𝐹𝑏) ⊆ 𝐵)
102, 9elpwd 4551 . . 3 (𝜑 → (𝐹𝑏) ∈ 𝒫 𝐵)
1110adantr 481 . 2 ((𝜑𝑏 ∈ 𝒫 𝐴) → (𝐹𝑏) ∈ 𝒫 𝐵)
12 f1opw2.2 . . . 4 (𝜑 → (𝐹𝑎) ∈ V)
13 imassrn 5998 . . . . 5 (𝐹𝑎) ⊆ ran 𝐹
14 dfdm4 5825 . . . . . 6 dom 𝐹 = ran 𝐹
15 f1odm 6758 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵 → dom 𝐹 = 𝐴)
164, 15syl 17 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
1714, 16eqtr3id 2791 . . . . 5 (𝜑 → ran 𝐹 = 𝐴)
1813, 17sseqtrid 3983 . . . 4 (𝜑 → (𝐹𝑎) ⊆ 𝐴)
1912, 18elpwd 4551 . . 3 (𝜑 → (𝐹𝑎) ∈ 𝒫 𝐴)
2019adantr 481 . 2 ((𝜑𝑎 ∈ 𝒫 𝐵) → (𝐹𝑎) ∈ 𝒫 𝐴)
21 elpwi 4552 . . . . . . 7 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
2221adantl 482 . . . . . 6 ((𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵) → 𝑎𝐵)
23 foimacnv 6771 . . . . . 6 ((𝐹:𝐴onto𝐵𝑎𝐵) → (𝐹 “ (𝐹𝑎)) = 𝑎)
246, 22, 23syl2an 596 . . . . 5 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝐹 “ (𝐹𝑎)) = 𝑎)
2524eqcomd 2743 . . . 4 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → 𝑎 = (𝐹 “ (𝐹𝑎)))
26 imaeq2 5983 . . . . 5 (𝑏 = (𝐹𝑎) → (𝐹𝑏) = (𝐹 “ (𝐹𝑎)))
2726eqeq2d 2748 . . . 4 (𝑏 = (𝐹𝑎) → (𝑎 = (𝐹𝑏) ↔ 𝑎 = (𝐹 “ (𝐹𝑎))))
2825, 27syl5ibrcom 246 . . 3 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (𝐹𝑎) → 𝑎 = (𝐹𝑏)))
29 f1of1 6753 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
304, 29syl 17 . . . . . 6 (𝜑𝐹:𝐴1-1𝐵)
31 elpwi 4552 . . . . . . 7 (𝑏 ∈ 𝒫 𝐴𝑏𝐴)
3231adantr 481 . . . . . 6 ((𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵) → 𝑏𝐴)
33 f1imacnv 6770 . . . . . 6 ((𝐹:𝐴1-1𝐵𝑏𝐴) → (𝐹 “ (𝐹𝑏)) = 𝑏)
3430, 32, 33syl2an 596 . . . . 5 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝐹 “ (𝐹𝑏)) = 𝑏)
3534eqcomd 2743 . . . 4 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → 𝑏 = (𝐹 “ (𝐹𝑏)))
36 imaeq2 5983 . . . . 5 (𝑎 = (𝐹𝑏) → (𝐹𝑎) = (𝐹 “ (𝐹𝑏)))
3736eqeq2d 2748 . . . 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 7565 1 (𝜑 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)):𝒫 𝐴1-1-onto→𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  Vcvv 3441  wss 3897  𝒫 cpw 4545  cmpt 5170  ccnv 5607  dom cdm 5608  ran crn 5609  cima 5611  1-1wf1 6463  ontowfo 6464  1-1-ontowf1o 6465
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pr 5367
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-br 5088  df-opab 5150  df-mpt 5171  df-id 5507  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-fun 6468  df-fn 6469  df-f 6470  df-f1 6471  df-fo 6472  df-f1o 6473
This theorem is referenced by:  f1opw  7567
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