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Theorem fopwdom 9119
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 6091 . . . . . 6 (𝐹𝑎) ⊆ ran 𝐹
2 dfdm4 5909 . . . . . . 7 dom 𝐹 = ran 𝐹
3 fof 6821 . . . . . . . 8 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
43fdmd 6747 . . . . . . 7 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
52, 4eqtr3id 2789 . . . . . 6 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐴)
61, 5sseqtrid 4048 . . . . 5 (𝐹:𝐴onto𝐵 → (𝐹𝑎) ⊆ 𝐴)
76adantl 481 . . . 4 ((𝐹𝑉𝐹:𝐴onto𝐵) → (𝐹𝑎) ⊆ 𝐴)
8 cnvexg 7947 . . . . . 6 (𝐹𝑉𝐹 ∈ V)
98adantr 480 . . . . 5 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝐹 ∈ V)
10 imaexg 7936 . . . . 5 (𝐹 ∈ V → (𝐹𝑎) ∈ V)
11 elpwg 4608 . . . . 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 6076 . . . . . . 7 ((𝐹𝑎) = (𝐹𝑏) → (𝐹 “ (𝐹𝑎)) = (𝐹 “ (𝐹𝑏)))
1615adantl 481 . . . . . 6 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹 “ (𝐹𝑎)) = (𝐹 “ (𝐹𝑏)))
17 simpllr 776 . . . . . . 7 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝐹:𝐴onto𝐵)
18 simplrl 777 . . . . . . . 8 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎 ∈ 𝒫 𝐵)
1918elpwid 4614 . . . . . . 7 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎𝐵)
20 foimacnv 6866 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑎𝐵) → (𝐹 “ (𝐹𝑎)) = 𝑎)
2117, 19, 20syl2anc 584 . . . . . 6 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹 “ (𝐹𝑎)) = 𝑎)
22 simplrr 778 . . . . . . . 8 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏 ∈ 𝒫 𝐵)
2322elpwid 4614 . . . . . . 7 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏𝐵)
24 foimacnv 6866 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑏𝐵) → (𝐹 “ (𝐹𝑏)) = 𝑏)
2517, 23, 24syl2anc 584 . . . . . 6 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹 “ (𝐹𝑏)) = 𝑏)
2616, 21, 253eqtr3d 2783 . . . . 5 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎 = 𝑏)
2726ex 412 . . . 4 (((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
28 imaeq2 6076 . . . 4 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
2927, 28impbid1 225 . . 3 (((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) → ((𝐹𝑎) = (𝐹𝑏) ↔ 𝑎 = 𝑏))
3029ex 412 . 2 ((𝐹𝑉𝐹:𝐴onto𝐵) → ((𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵) → ((𝐹𝑎) = (𝐹𝑏) ↔ 𝑎 = 𝑏)))
31 rnexg 7925 . . . . 5 (𝐹𝑉 → ran 𝐹 ∈ V)
32 forn 6824 . . . . . 6 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
3332eleq1d 2824 . . . . 5 (𝐹:𝐴onto𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
3431, 33syl5ibcom 245 . . . 4 (𝐹𝑉 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
3534imp 406 . . 3 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝐵 ∈ V)
3635pwexd 5385 . 2 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝒫 𝐵 ∈ V)
37 dmfex 7928 . . . 4 ((𝐹𝑉𝐹:𝐴𝐵) → 𝐴 ∈ V)
383, 37sylan2 593 . . 3 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝐴 ∈ V)
3938pwexd 5385 . 2 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝒫 𝐴 ∈ V)
4014, 30, 36, 39dom3d 9033 1 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963  𝒫 cpw 4605   class class class wbr 5148  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  wf 6559  ontowfo 6561  cdom 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-fv 6571  df-dom 8986
This theorem is referenced by:  pwdom  9168  wdompwdom  9616
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