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Theorem f1opw2 7005
Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 7006 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 2724 . 2 (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)) = (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏))
2 imassrn 5587 . . . . 5 (𝐹𝑏) ⊆ ran 𝐹
3 f1opw2.1 . . . . . . 7 (𝜑𝐹:𝐴1-1-onto𝐵)
4 f1ofo 6257 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
53, 4syl 17 . . . . . 6 (𝜑𝐹:𝐴onto𝐵)
6 forn 6231 . . . . . 6 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
75, 6syl 17 . . . . 5 (𝜑 → ran 𝐹 = 𝐵)
82, 7syl5sseq 3759 . . . 4 (𝜑 → (𝐹𝑏) ⊆ 𝐵)
9 f1opw2.3 . . . . 5 (𝜑 → (𝐹𝑏) ∈ V)
10 elpwg 4274 . . . . 5 ((𝐹𝑏) ∈ V → ((𝐹𝑏) ∈ 𝒫 𝐵 ↔ (𝐹𝑏) ⊆ 𝐵))
119, 10syl 17 . . . 4 (𝜑 → ((𝐹𝑏) ∈ 𝒫 𝐵 ↔ (𝐹𝑏) ⊆ 𝐵))
128, 11mpbird 247 . . 3 (𝜑 → (𝐹𝑏) ∈ 𝒫 𝐵)
1312adantr 472 . 2 ((𝜑𝑏 ∈ 𝒫 𝐴) → (𝐹𝑏) ∈ 𝒫 𝐵)
14 imassrn 5587 . . . . 5 (𝐹𝑎) ⊆ ran 𝐹
15 dfdm4 5423 . . . . . 6 dom 𝐹 = ran 𝐹
16 f1odm 6254 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵 → dom 𝐹 = 𝐴)
173, 16syl 17 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
1815, 17syl5eqr 2772 . . . . 5 (𝜑 → ran 𝐹 = 𝐴)
1914, 18syl5sseq 3759 . . . 4 (𝜑 → (𝐹𝑎) ⊆ 𝐴)
20 f1opw2.2 . . . . 5 (𝜑 → (𝐹𝑎) ∈ V)
21 elpwg 4274 . . . . 5 ((𝐹𝑎) ∈ V → ((𝐹𝑎) ∈ 𝒫 𝐴 ↔ (𝐹𝑎) ⊆ 𝐴))
2220, 21syl 17 . . . 4 (𝜑 → ((𝐹𝑎) ∈ 𝒫 𝐴 ↔ (𝐹𝑎) ⊆ 𝐴))
2319, 22mpbird 247 . . 3 (𝜑 → (𝐹𝑎) ∈ 𝒫 𝐴)
2423adantr 472 . 2 ((𝜑𝑎 ∈ 𝒫 𝐵) → (𝐹𝑎) ∈ 𝒫 𝐴)
25 elpwi 4276 . . . . . . 7 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
2625adantl 473 . . . . . 6 ((𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵) → 𝑎𝐵)
27 foimacnv 6267 . . . . . 6 ((𝐹:𝐴onto𝐵𝑎𝐵) → (𝐹 “ (𝐹𝑎)) = 𝑎)
285, 26, 27syl2an 495 . . . . 5 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝐹 “ (𝐹𝑎)) = 𝑎)
2928eqcomd 2730 . . . 4 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → 𝑎 = (𝐹 “ (𝐹𝑎)))
30 imaeq2 5572 . . . . 5 (𝑏 = (𝐹𝑎) → (𝐹𝑏) = (𝐹 “ (𝐹𝑎)))
3130eqeq2d 2734 . . . 4 (𝑏 = (𝐹𝑎) → (𝑎 = (𝐹𝑏) ↔ 𝑎 = (𝐹 “ (𝐹𝑎))))
3229, 31syl5ibrcom 237 . . 3 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (𝐹𝑎) → 𝑎 = (𝐹𝑏)))
33 f1of1 6249 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
343, 33syl 17 . . . . . 6 (𝜑𝐹:𝐴1-1𝐵)
35 elpwi 4276 . . . . . . 7 (𝑏 ∈ 𝒫 𝐴𝑏𝐴)
3635adantr 472 . . . . . 6 ((𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵) → 𝑏𝐴)
37 f1imacnv 6266 . . . . . 6 ((𝐹:𝐴1-1𝐵𝑏𝐴) → (𝐹 “ (𝐹𝑏)) = 𝑏)
3834, 36, 37syl2an 495 . . . . 5 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝐹 “ (𝐹𝑏)) = 𝑏)
3938eqcomd 2730 . . . 4 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → 𝑏 = (𝐹 “ (𝐹𝑏)))
40 imaeq2 5572 . . . . 5 (𝑎 = (𝐹𝑏) → (𝐹𝑎) = (𝐹 “ (𝐹𝑏)))
4140eqeq2d 2734 . . . 4 (𝑎 = (𝐹𝑏) → (𝑏 = (𝐹𝑎) ↔ 𝑏 = (𝐹 “ (𝐹𝑏))))
4239, 41syl5ibrcom 237 . . 3 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝑎 = (𝐹𝑏) → 𝑏 = (𝐹𝑎)))
4332, 42impbid 202 . 2 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (𝐹𝑎) ↔ 𝑎 = (𝐹𝑏)))
441, 13, 24, 43f1o2d 7004 1 (𝜑 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)):𝒫 𝐴1-1-onto→𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  Vcvv 3304  wss 3680  𝒫 cpw 4266  cmpt 4837  ccnv 5217  dom cdm 5218  ran crn 5219  cima 5221  1-1wf1 5998  ontowfo 5999  1-1-ontowf1o 6000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008
This theorem is referenced by:  f1opw  7006
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