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Theorem fopwdom 9079
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 6063 . . . . . 6 (𝐹𝑎) ⊆ ran 𝐹
2 dfdm4 5888 . . . . . . 7 dom 𝐹 = ran 𝐹
3 fof 6798 . . . . . . . 8 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
43fdmd 6721 . . . . . . 7 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
52, 4eqtr3id 2780 . . . . . 6 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐴)
61, 5sseqtrid 4029 . . . . 5 (𝐹:𝐴onto𝐵 → (𝐹𝑎) ⊆ 𝐴)
76adantl 481 . . . 4 ((𝐹𝑉𝐹:𝐴onto𝐵) → (𝐹𝑎) ⊆ 𝐴)
8 cnvexg 7911 . . . . . 6 (𝐹𝑉𝐹 ∈ V)
98adantr 480 . . . . 5 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝐹 ∈ V)
10 imaexg 7902 . . . . 5 (𝐹 ∈ V → (𝐹𝑎) ∈ V)
11 elpwg 4600 . . . . 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 6048 . . . . . . 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 773 . . . . . . 7 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝐹:𝐴onto𝐵)
18 simplrl 774 . . . . . . . 8 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎 ∈ 𝒫 𝐵)
1918elpwid 4606 . . . . . . 7 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎𝐵)
20 foimacnv 6843 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑎𝐵) → (𝐹 “ (𝐹𝑎)) = 𝑎)
2117, 19, 20syl2anc 583 . . . . . 6 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹 “ (𝐹𝑎)) = 𝑎)
22 simplrr 775 . . . . . . . 8 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏 ∈ 𝒫 𝐵)
2322elpwid 4606 . . . . . . 7 ((((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏𝐵)
24 foimacnv 6843 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑏𝐵) → (𝐹 “ (𝐹𝑏)) = 𝑏)
2517, 23, 24syl2anc 583 . . . . . 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 412 . . . 4 (((𝐹𝑉𝐹:𝐴onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵𝑏 ∈ 𝒫 𝐵)) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
28 imaeq2 6048 . . . 4 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
2927, 28impbid1 224 . . 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 7891 . . . . 5 (𝐹𝑉 → ran 𝐹 ∈ V)
32 forn 6801 . . . . . 6 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
3332eleq1d 2812 . . . . 5 (𝐹:𝐴onto𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
3431, 33syl5ibcom 244 . . . 4 (𝐹𝑉 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
3534imp 406 . . 3 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝐵 ∈ V)
3635pwexd 5370 . 2 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝒫 𝐵 ∈ V)
37 dmfex 7894 . . . 4 ((𝐹𝑉𝐹:𝐴𝐵) → 𝐴 ∈ V)
383, 37sylan2 592 . . 3 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝐴 ∈ V)
3938pwexd 5370 . 2 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝒫 𝐴 ∈ V)
4014, 30, 36, 39dom3d 8989 1 ((𝐹𝑉𝐹:𝐴onto𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  wss 3943  𝒫 cpw 4597   class class class wbr 5141  ccnv 5668  dom cdm 5669  ran crn 5670  cima 5672  wf 6532  ontowfo 6534  cdom 8936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-fv 6544  df-dom 8940
This theorem is referenced by:  pwdom  9128  wdompwdom  9572
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