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Theorem en3d 8549
Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
Hypotheses
Ref Expression
en3d.1 (𝜑𝐴 ∈ V)
en3d.2 (𝜑𝐵 ∈ V)
en3d.3 (𝜑 → (𝑥𝐴𝐶𝐵))
en3d.4 (𝜑 → (𝑦𝐵𝐷𝐴))
en3d.5 (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶)))
Assertion
Ref Expression
en3d (𝜑𝐴𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem en3d
StepHypRef Expression
1 en3d.1 . 2 (𝜑𝐴 ∈ V)
2 en3d.2 . 2 (𝜑𝐵 ∈ V)
3 eqid 2824 . . 3 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
4 en3d.3 . . . 4 (𝜑 → (𝑥𝐴𝐶𝐵))
54imp 409 . . 3 ((𝜑𝑥𝐴) → 𝐶𝐵)
6 en3d.4 . . . 4 (𝜑 → (𝑦𝐵𝐷𝐴))
76imp 409 . . 3 ((𝜑𝑦𝐵) → 𝐷𝐴)
8 en3d.5 . . . 4 (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶)))
98imp 409 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))
103, 5, 7, 9f1o2d 7402 . 2 (𝜑 → (𝑥𝐴𝐶):𝐴1-1-onto𝐵)
11 f1oen2g 8529 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ (𝑥𝐴𝐶):𝐴1-1-onto𝐵) → 𝐴𝐵)
121, 2, 10, 11syl3anc 1367 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1536  wcel 2113  Vcvv 3497   class class class wbr 5069  cmpt 5149  1-1-ontowf1o 6357  cen 8509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-en 8513
This theorem is referenced by:  en3i  8551  fundmen  8586  mapen  8684  mapxpen  8686  mapunen  8689  ssenen  8694  fzen  12927  hashbclem  13813  hashfacen  13815  hashf1lem1  13816  hashdvds  16115  sylow2a  18747  lsmhash  18834  subfacp1lem3  32433  subfacp1lem5  32435
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