ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  en3d GIF version

Theorem en3d 6249
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 2040 . . 3 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
4 en3d.3 . . . 4 (𝜑 → (𝑥𝐴𝐶𝐵))
54imp 115 . . 3 ((𝜑𝑥𝐴) → 𝐶𝐵)
6 en3d.4 . . . 4 (𝜑 → (𝑦𝐵𝐷𝐴))
76imp 115 . . 3 ((𝜑𝑦𝐵) → 𝐷𝐴)
8 en3d.5 . . . 4 (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶)))
98imp 115 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))
103, 5, 7, 9f1o2d 5705 . 2 (𝜑 → (𝑥𝐴𝐶):𝐴1-1-onto𝐵)
11 f1oen2g 6235 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ (𝑥𝐴𝐶):𝐴1-1-onto𝐵) → 𝐴𝐵)
121, 2, 10, 11syl3anc 1135 1 (𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  Vcvv 2557   class class class wbr 3764  cmpt 3818  1-1-ontowf1o 4901  cen 6219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-en 6222
This theorem is referenced by:  en3i  6251  fundmen  6286  fzen  8907
  Copyright terms: Public domain W3C validator