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Theorem en3d 8985
Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by AV, 4-Aug-2024.)
Hypotheses
Ref Expression
en3d.1 (𝜑𝐴𝑉)
en3d.2 (𝜑𝐵𝑊)
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)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem en3d
StepHypRef Expression
1 en3d.1 . 2 (𝜑𝐴𝑉)
2 en3d.2 . 2 (𝜑𝐵𝑊)
3 eqid 2733 . . 3 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
4 en3d.3 . . . 4 (𝜑 → (𝑥𝐴𝐶𝐵))
54imp 408 . . 3 ((𝜑𝑥𝐴) → 𝐶𝐵)
6 en3d.4 . . . 4 (𝜑 → (𝑦𝐵𝐷𝐴))
76imp 408 . . 3 ((𝜑𝑦𝐵) → 𝐷𝐴)
8 en3d.5 . . . 4 (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶)))
98imp 408 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))
103, 5, 7, 9f1o2d 7660 . 2 (𝜑 → (𝑥𝐴𝐶):𝐴1-1-onto𝐵)
11 f1oen2g 8964 . 2 ((𝐴𝑉𝐵𝑊 ∧ (𝑥𝐴𝐶):𝐴1-1-onto𝐵) → 𝐴𝐵)
121, 2, 10, 11syl3anc 1372 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107   class class class wbr 5149  cmpt 5232  1-1-ontowf1o 6543  cen 8936
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 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-en 8940
This theorem is referenced by:  en3i  8987  fundmen  9031  mapen  9141  mapxpen  9143  mapunen  9146  ssenen  9151  fzen  13518  hashbclem  14411  hashfacen  14413  hashfacenOLD  14414  hashf1lem1  14415  hashf1lem1OLD  14416  hashdvds  16708
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