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Theorem en3d 7936
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 2621 . . 3 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
4 en3d.3 . . . 4 (𝜑 → (𝑥𝐴𝐶𝐵))
54imp 445 . . 3 ((𝜑𝑥𝐴) → 𝐶𝐵)
6 en3d.4 . . . 4 (𝜑 → (𝑦𝐵𝐷𝐴))
76imp 445 . . 3 ((𝜑𝑦𝐵) → 𝐷𝐴)
8 en3d.5 . . . 4 (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶)))
98imp 445 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))
103, 5, 7, 9f1o2d 6840 . 2 (𝜑 → (𝑥𝐴𝐶):𝐴1-1-onto𝐵)
11 f1oen2g 7916 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ (𝑥𝐴𝐶):𝐴1-1-onto𝐵) → 𝐴𝐵)
121, 2, 10, 11syl3anc 1323 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3186   class class class wbr 4613  cmpt 4673  1-1-ontowf1o 5846  cen 7896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-en 7900
This theorem is referenced by:  en3i  7938  fundmen  7974  mapen  8068  mapxpen  8070  mapunen  8073  ssenen  8078  fzen  12300  hashbclem  13174  hashfacen  13176  hashf1lem1  13177  hashdvds  15404  sylow2a  17955  lsmhash  18039  subfacp1lem3  30872  subfacp1lem5  30874
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