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Theorem fopwdom 9120
Description: Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.) (Revised by AV, 18-Sep-2021.)
Assertion
Ref Expression
fopwdom ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)

Proof of Theorem fopwdom
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imassrn 6082 . . . . . 6 (𝐹𝑎) ⊆ ran 𝐹
2 dfdm4 5904 . . . . . . 7 dom 𝐹 = ran 𝐹
3 fof 6817 . . . . . . . 8 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
43fdmd 6740 . . . . . . 7 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
52, 4eqtr3id 2780 . . . . . 6 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐴)
61, 5sseqtrid 4032 . . . . 5 (𝐹:𝐴onto𝐵 → (𝐹𝑎) ⊆ 𝐴)
76adantl 480 . . . 4 ((𝐹𝑉𝐹:𝐴onto𝐵) → (𝐹𝑎) ⊆ 𝐴)
8 cnvexg 7939 . . . . . 6 (𝐹𝑉𝐹 ∈ V)
98adantr 479 . . . . 5 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝐹 ∈ V)
10 imaexg 7928 . . . . 5 (𝐹 ∈ V → (𝐹𝑎) ∈ V)
11 elpwg 4610 . . . . 5 ((𝐹𝑎) ∈ V → ((𝐹𝑎) ∈ 𝒫 𝐴 ↔ (𝐹𝑎) ⊆ 𝐴))
129, 10, 113syl 18 . . . 4 ((𝐹𝑉𝐹:𝐴onto𝐵) → ((𝐹𝑎) ∈ 𝒫 𝐴 ↔ (𝐹𝑎) ⊆ 𝐴))
137, 12mpbird 256 . . 3 ((𝐹𝑉𝐹:𝐴onto𝐵) → (𝐹𝑎) ∈ 𝒫 𝐴)
1413a1d 25 . 2 ((𝐹𝑉𝐹:𝐴onto𝐵) → (𝑎 ∈ 𝒫 𝐵 → (𝐹𝑎) ∈ 𝒫 𝐴))
15 imaeq2 6067 . . . . . . 7 ((𝐹𝑎) = (𝐹𝑏) → (𝐹 “ (𝐹𝑎)) = (𝐹 “ (𝐹𝑏)))
1615adantl 480 . . . . . 6 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹 “ (𝐹𝑎)) = (𝐹 “ (𝐹𝑏)))
17 simpllr 774 . . . . . . 7 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝐹:𝐴onto𝐵)
18 simplrl 775 . . . . . . . 8 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎 ∈ 𝒫 𝐵)
1918elpwid 4616 . . . . . . 7 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎𝐵)
20 foimacnv 6862 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑎𝐵) → (𝐹 “ (𝐹𝑎)) = 𝑎)
2117, 19, 20syl2anc 582 . . . . . 6 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹 “ (𝐹𝑎)) = 𝑎)
22 simplrr 776 . . . . . . . 8 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏 ∈ 𝒫 𝐵)
2322elpwid 4616 . . . . . . 7 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏𝐵)
24 foimacnv 6862 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑏𝐵) → (𝐹 “ (𝐹𝑏)) = 𝑏)
2517, 23, 24syl2anc 582 . . . . . 6 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹 “ (𝐹𝑏)) = 𝑏)
2616, 21, 253eqtr3d 2774 . . . . 5 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎 = 𝑏)
2726ex 411 . . . 4 (((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
28 imaeq2 6067 . . . 4 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
2927, 28impbid1 224 . . 3 (((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) → ((𝐹𝑎) = (𝐹𝑏) ↔ 𝑎 = 𝑏))
3029ex 411 . 2 ((𝐹𝑉𝐹:𝐴onto𝐵) → ((𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵) → ((𝐹𝑎) = (𝐹𝑏) ↔ 𝑎 = 𝑏)))
31 rnexg 7917 . . . . 5 (𝐹𝑉 → ran 𝐹 ∈ V)
32 forn 6820 . . . . . 6 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
3332eleq1d 2811 . . . . 5 (𝐹:𝐴onto𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
3431, 33syl5ibcom 244 . . . 4 (𝐹𝑉 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
3534imp 405 . . 3 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝐵 ∈ V)
3635pwexd 5385 . 2 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝒫 𝐵 ∈ V)
37 dmfex 7920 . . . 4 ((𝐹𝑉𝐹:𝐴𝐵) → 𝐴 ∈ V)
383, 37sylan2 591 . . 3 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝐴 ∈ V)
3938pwexd 5385 . 2 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝒫 𝐴 ∈ V)
4014, 30, 36, 39dom3d 9027 1 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  Vcvv 3462  wss 3947  𝒫 cpw 4607   class class class wbr 5155  ccnv 5683  dom cdm 5684  ran crn 5685  cima 5687  wf 6552  ontowfo 6554  cdom 8974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-fv 6564  df-dom 8978
This theorem is referenced by:  pwdom  9169  wdompwdom  9623
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