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Theorem fopwdom 9031
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 6029 . . . . . 6 (𝐹𝑎) ⊆ ran 𝐹
2 dfdm4 5856 . . . . . . 7 dom 𝐹 = ran 𝐹
3 fof 6761 . . . . . . . 8 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
43fdmd 6684 . . . . . . 7 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
52, 4eqtr3id 2791 . . . . . 6 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐴)
61, 5sseqtrid 4001 . . . . 5 (𝐹:𝐴onto𝐵 → (𝐹𝑎) ⊆ 𝐴)
76adantl 483 . . . 4 ((𝐹𝑉𝐹:𝐴onto𝐵) → (𝐹𝑎) ⊆ 𝐴)
8 cnvexg 7866 . . . . . 6 (𝐹𝑉𝐹 ∈ V)
98adantr 482 . . . . 5 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝐹 ∈ V)
10 imaexg 7857 . . . . 5 (𝐹 ∈ V → (𝐹𝑎) ∈ V)
11 elpwg 4568 . . . . 5 ((𝐹𝑎) ∈ V → ((𝐹𝑎) ∈ 𝒫 𝐴 ↔ (𝐹𝑎) ⊆ 𝐴))
129, 10, 113syl 18 . . . 4 ((𝐹𝑉𝐹:𝐴onto𝐵) → ((𝐹𝑎) ∈ 𝒫 𝐴 ↔ (𝐹𝑎) ⊆ 𝐴))
137, 12mpbird 257 . . 3 ((𝐹𝑉𝐹:𝐴onto𝐵) → (𝐹𝑎) ∈ 𝒫 𝐴)
1413a1d 25 . 2 ((𝐹𝑉𝐹:𝐴onto𝐵) → (𝑎 ∈ 𝒫 𝐵 → (𝐹𝑎) ∈ 𝒫 𝐴))
15 imaeq2 6014 . . . . . . 7 ((𝐹𝑎) = (𝐹𝑏) → (𝐹 “ (𝐹𝑎)) = (𝐹 “ (𝐹𝑏)))
1615adantl 483 . . . . . 6 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹 “ (𝐹𝑎)) = (𝐹 “ (𝐹𝑏)))
17 simpllr 775 . . . . . . 7 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝐹:𝐴onto𝐵)
18 simplrl 776 . . . . . . . 8 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎 ∈ 𝒫 𝐵)
1918elpwid 4574 . . . . . . 7 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎𝐵)
20 foimacnv 6806 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑎𝐵) → (𝐹 “ (𝐹𝑎)) = 𝑎)
2117, 19, 20syl2anc 585 . . . . . 6 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹 “ (𝐹𝑎)) = 𝑎)
22 simplrr 777 . . . . . . . 8 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏 ∈ 𝒫 𝐵)
2322elpwid 4574 . . . . . . 7 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏𝐵)
24 foimacnv 6806 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑏𝐵) → (𝐹 “ (𝐹𝑏)) = 𝑏)
2517, 23, 24syl2anc 585 . . . . . 6 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹 “ (𝐹𝑏)) = 𝑏)
2616, 21, 253eqtr3d 2785 . . . . 5 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎 = 𝑏)
2726ex 414 . . . 4 (((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
28 imaeq2 6014 . . . 4 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
2927, 28impbid1 224 . . 3 (((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) → ((𝐹𝑎) = (𝐹𝑏) ↔ 𝑎 = 𝑏))
3029ex 414 . 2 ((𝐹𝑉𝐹:𝐴onto𝐵) → ((𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵) → ((𝐹𝑎) = (𝐹𝑏) ↔ 𝑎 = 𝑏)))
31 rnexg 7846 . . . . 5 (𝐹𝑉 → ran 𝐹 ∈ V)
32 forn 6764 . . . . . 6 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
3332eleq1d 2823 . . . . 5 (𝐹:𝐴onto𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
3431, 33syl5ibcom 244 . . . 4 (𝐹𝑉 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
3534imp 408 . . 3 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝐵 ∈ V)
3635pwexd 5339 . 2 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝒫 𝐵 ∈ V)
37 dmfex 7849 . . . 4 ((𝐹𝑉𝐹:𝐴𝐵) → 𝐴 ∈ V)
383, 37sylan2 594 . . 3 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝐴 ∈ V)
3938pwexd 5339 . 2 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝒫 𝐴 ∈ V)
4014, 30, 36, 39dom3d 8941 1 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3448  wss 3915  𝒫 cpw 4565   class class class wbr 5110  ccnv 5637  dom cdm 5638  ran crn 5639  cima 5641  wf 6497  ontowfo 6499  cdom 8888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-fv 6509  df-dom 8892
This theorem is referenced by:  pwdom  9080  wdompwdom  9521
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