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Theorem f1opw2 7384
Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 7385 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 2801 . 2 (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)) = (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏))
2 f1opw2.3 . . . 4 (𝜑 → (𝐹𝑏) ∈ V)
3 imassrn 5911 . . . . 5 (𝐹𝑏) ⊆ ran 𝐹
4 f1opw2.1 . . . . . . 7 (𝜑𝐹:𝐴1-1-onto𝐵)
5 f1ofo 6601 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
64, 5syl 17 . . . . . 6 (𝜑𝐹:𝐴onto𝐵)
7 forn 6572 . . . . . 6 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
86, 7syl 17 . . . . 5 (𝜑 → ran 𝐹 = 𝐵)
93, 8sseqtrid 3970 . . . 4 (𝜑 → (𝐹𝑏) ⊆ 𝐵)
102, 9elpwd 4508 . . 3 (𝜑 → (𝐹𝑏) ∈ 𝒫 𝐵)
1110adantr 484 . 2 ((𝜑𝑏 ∈ 𝒫 𝐴) → (𝐹𝑏) ∈ 𝒫 𝐵)
12 f1opw2.2 . . . 4 (𝜑 → (𝐹𝑎) ∈ V)
13 imassrn 5911 . . . . 5 (𝐹𝑎) ⊆ ran 𝐹
14 dfdm4 5732 . . . . . 6 dom 𝐹 = ran 𝐹
15 f1odm 6598 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵 → dom 𝐹 = 𝐴)
164, 15syl 17 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
1714, 16syl5eqr 2850 . . . . 5 (𝜑 → ran 𝐹 = 𝐴)
1813, 17sseqtrid 3970 . . . 4 (𝜑 → (𝐹𝑎) ⊆ 𝐴)
1912, 18elpwd 4508 . . 3 (𝜑 → (𝐹𝑎) ∈ 𝒫 𝐴)
2019adantr 484 . 2 ((𝜑𝑎 ∈ 𝒫 𝐵) → (𝐹𝑎) ∈ 𝒫 𝐴)
21 elpwi 4509 . . . . . . 7 (𝑎 ∈ 𝒫 𝐵𝑎𝐵)
2221adantl 485 . . . . . 6 ((𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵) → 𝑎𝐵)
23 foimacnv 6611 . . . . . 6 ((𝐹:𝐴onto𝐵𝑎𝐵) → (𝐹 “ (𝐹𝑎)) = 𝑎)
246, 22, 23syl2an 598 . . . . 5 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝐹 “ (𝐹𝑎)) = 𝑎)
2524eqcomd 2807 . . . 4 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → 𝑎 = (𝐹 “ (𝐹𝑎)))
26 imaeq2 5896 . . . . 5 (𝑏 = (𝐹𝑎) → (𝐹𝑏) = (𝐹 “ (𝐹𝑎)))
2726eqeq2d 2812 . . . 4 (𝑏 = (𝐹𝑎) → (𝑎 = (𝐹𝑏) ↔ 𝑎 = (𝐹 “ (𝐹𝑎))))
2825, 27syl5ibrcom 250 . . 3 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (𝐹𝑎) → 𝑎 = (𝐹𝑏)))
29 f1of1 6593 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
304, 29syl 17 . . . . . 6 (𝜑𝐹:𝐴1-1𝐵)
31 elpwi 4509 . . . . . . 7 (𝑏 ∈ 𝒫 𝐴𝑏𝐴)
3231adantr 484 . . . . . 6 ((𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵) → 𝑏𝐴)
33 f1imacnv 6610 . . . . . 6 ((𝐹:𝐴1-1𝐵𝑏𝐴) → (𝐹 “ (𝐹𝑏)) = 𝑏)
3430, 32, 33syl2an 598 . . . . 5 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝐹 “ (𝐹𝑏)) = 𝑏)
3534eqcomd 2807 . . . 4 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → 𝑏 = (𝐹 “ (𝐹𝑏)))
36 imaeq2 5896 . . . . 5 (𝑎 = (𝐹𝑏) → (𝐹𝑎) = (𝐹 “ (𝐹𝑏)))
3736eqeq2d 2812 . . . 4 (𝑎 = (𝐹𝑏) → (𝑏 = (𝐹𝑎) ↔ 𝑏 = (𝐹 “ (𝐹𝑏))))
3835, 37syl5ibrcom 250 . . 3 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝑎 = (𝐹𝑏) → 𝑏 = (𝐹𝑎)))
3928, 38impbid 215 . 2 ((𝜑 ∧ (𝑏 ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (𝐹𝑎) ↔ 𝑎 = (𝐹𝑏)))
401, 11, 20, 39f1o2d 7383 1 (𝜑 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)):𝒫 𝐴1-1-onto→𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  Vcvv 3444  wss 3884  𝒫 cpw 4500  cmpt 5113  ccnv 5522  dom cdm 5523  ran crn 5524  cima 5526  1-1wf1 6325  ontowfo 6326  1-1-ontowf1o 6327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335
This theorem is referenced by:  f1opw  7385
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