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Theorem f1opw2 5706
 Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 5707 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 2040 . 2 (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)) = (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏))
2 imassrn 4679 . . . . 5 (𝐹𝑏) ⊆ ran 𝐹
3 f1opw2.1 . . . . . . 7 (𝜑𝐹:𝐴1-1-onto𝐵)
4 f1ofo 5133 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
53, 4syl 14 . . . . . 6 (𝜑𝐹:𝐴onto𝐵)
6 forn 5109 . . . . . 6 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
75, 6syl 14 . . . . 5 (𝜑 → ran 𝐹 = 𝐵)
82, 7syl5sseq 2993 . . . 4 (𝜑 → (𝐹𝑏) ⊆ 𝐵)
9 f1opw2.3 . . . . 5 (𝜑 → (𝐹𝑏) ∈ V)
10 elpwg 3367 . . . . 5 ((𝐹𝑏) ∈ V → ((𝐹𝑏) ∈ 𝒫 𝐵 ↔ (𝐹𝑏) ⊆ 𝐵))
119, 10syl 14 . . . 4 (𝜑 → ((𝐹𝑏) ∈ 𝒫 𝐵 ↔ (𝐹𝑏) ⊆ 𝐵))
128, 11mpbird 156 . . 3 (𝜑 → (𝐹𝑏) ∈ 𝒫 𝐵)
1312adantr 261 . 2 ((𝜑𝑏 ∈ 𝒫 𝐴) → (𝐹𝑏) ∈ 𝒫 𝐵)
14 imassrn 4679 . . . . 5 (𝐹𝑎) ⊆ ran 𝐹
15 dfdm4 4527 . . . . . 6 dom 𝐹 = ran 𝐹
16 f1odm 5130 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵 → dom 𝐹 = 𝐴)
173, 16syl 14 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
1815, 17syl5eqr 2086 . . . . 5 (𝜑 → ran 𝐹 = 𝐴)
1914, 18syl5sseq 2993 . . . 4 (𝜑 → (𝐹𝑎) ⊆ 𝐴)
20 f1opw2.2 . . . . 5 (𝜑 → (𝐹𝑎) ∈ V)
21 elpwg 3367 . . . . 5 ((𝐹𝑎) ∈ V → ((𝐹𝑎) ∈ 𝒫 𝐴 ↔ (𝐹𝑎) ⊆ 𝐴))
2220, 21syl 14 . . . 4 (𝜑 → ((𝐹𝑎) ∈ 𝒫 𝐴 ↔ (𝐹𝑎) ⊆ 𝐴))
2319, 22mpbird 156 . . 3 (𝜑 → (𝐹𝑎) ∈ 𝒫 𝐴)
2423adantr 261 . 2 ((𝜑𝑎 ∈ 𝒫 𝐵) → (𝐹𝑎) ∈ 𝒫 𝐴)
25 elpwi 3368 . . . . . . 7 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
2625adantl 262 . . . . . 6 ((𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵) → 𝑎𝐵)
27 foimacnv 5144 . . . . . 6 ((𝐹:𝐴onto𝐵𝑎𝐵) → (𝐹 “ (𝐹𝑎)) = 𝑎)
285, 26, 27syl2an 273 . . . . 5 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝐹 “ (𝐹𝑎)) = 𝑎)
2928eqcomd 2045 . . . 4 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → 𝑎 = (𝐹 “ (𝐹𝑎)))
30 imaeq2 4664 . . . . 5 (𝑏 = (𝐹𝑎) → (𝐹𝑏) = (𝐹 “ (𝐹𝑎)))
3130eqeq2d 2051 . . . 4 (𝑏 = (𝐹𝑎) → (𝑎 = (𝐹𝑏) ↔ 𝑎 = (𝐹 “ (𝐹𝑎))))
3229, 31syl5ibrcom 146 . . 3 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (𝐹𝑎) → 𝑎 = (𝐹𝑏)))
33 f1of1 5125 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
343, 33syl 14 . . . . . 6 (𝜑𝐹:𝐴1-1𝐵)
35 elpwi 3368 . . . . . . 7 (𝑏 ∈ 𝒫 𝐴𝑏𝐴)
3635adantr 261 . . . . . 6 ((𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵) → 𝑏𝐴)
37 f1imacnv 5143 . . . . . 6 ((𝐹:𝐴1-1𝐵𝑏𝐴) → (𝐹 “ (𝐹𝑏)) = 𝑏)
3834, 36, 37syl2an 273 . . . . 5 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝐹 “ (𝐹𝑏)) = 𝑏)
3938eqcomd 2045 . . . 4 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → 𝑏 = (𝐹 “ (𝐹𝑏)))
40 imaeq2 4664 . . . . 5 (𝑎 = (𝐹𝑏) → (𝐹𝑎) = (𝐹 “ (𝐹𝑏)))
4140eqeq2d 2051 . . . 4 (𝑎 = (𝐹𝑏) → (𝑏 = (𝐹𝑎) ↔ 𝑏 = (𝐹 “ (𝐹𝑏))))
4239, 41syl5ibrcom 146 . . 3 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝑎 = (𝐹𝑏) → 𝑏 = (𝐹𝑎)))
4332, 42impbid 120 . 2 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (𝐹𝑎) ↔ 𝑎 = (𝐹𝑏)))
441, 13, 24, 43f1o2d 5705 1 (𝜑 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)):𝒫 𝐴1-1-onto→𝒫 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  𝒫 cpw 3359   ↦ cmpt 3818  ◡ccnv 4344  dom cdm 4345  ran crn 4346   “ cima 4348  –1-1→wf1 4899  –onto→wfo 4900  –1-1-onto→wf1o 4901 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909 This theorem is referenced by:  f1opw  5707
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