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Theorem en3d 6185
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 → (x A𝐶 B))
en3d.4 (φ → (y B𝐷 A))
en3d.5 (φ → ((x A y B) → (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 2037 . . 3 (x A𝐶) = (x A𝐶)
4 en3d.3 . . . 4 (φ → (x A𝐶 B))
54imp 115 . . 3 ((φ x A) → 𝐶 B)
6 en3d.4 . . . 4 (φ → (y B𝐷 A))
76imp 115 . . 3 ((φ y B) → 𝐷 A)
8 en3d.5 . . . 4 (φ → ((x A y B) → (x = 𝐷y = 𝐶)))
98imp 115 . . 3 ((φ (x A y B)) → (x = 𝐷y = 𝐶))
103, 5, 7, 9f1o2d 5647 . 2 (φ → (x A𝐶):A1-1-ontoB)
11 f1oen2g 6171 . 2 ((A V B V (x A𝐶):A1-1-ontoB) → AB)
121, 2, 10, 11syl3anc 1134 1 (φAB)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  Vcvv 2551   class class class wbr 3755  cmpt 3809  1-1-ontowf1o 4844  cen 6155
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-en 6158
This theorem is referenced by:  en3i  6187  fundmen  6222  fzen  8657
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