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Theorem en2d 6669
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
en2d.1 (𝜑𝐴 ∈ V)
en2d.2 (𝜑𝐵 ∈ V)
en2d.3 (𝜑 → (𝑥𝐴𝐶 ∈ V))
en2d.4 (𝜑 → (𝑦𝐵𝐷 ∈ V))
en2d.5 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
Assertion
Ref Expression
en2d (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 en2d
StepHypRef Expression
1 en2d.1 . 2 (𝜑𝐴 ∈ V)
2 en2d.2 . 2 (𝜑𝐵 ∈ V)
3 eqid 2140 . . 3 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
4 en2d.3 . . . 4 (𝜑 → (𝑥𝐴𝐶 ∈ V))
54imp 123 . . 3 ((𝜑𝑥𝐴) → 𝐶 ∈ V)
6 en2d.4 . . . 4 (𝜑 → (𝑦𝐵𝐷 ∈ V))
76imp 123 . . 3 ((𝜑𝑦𝐵) → 𝐷 ∈ V)
8 en2d.5 . . 3 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
93, 5, 7, 8f1od 5980 . 2 (𝜑 → (𝑥𝐴𝐶):𝐴1-1-onto𝐵)
10 f1oen2g 6656 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ (𝑥𝐴𝐶):𝐴1-1-onto𝐵) → 𝐴𝐵)
111, 2, 9, 10syl3anc 1217 1 (𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  Vcvv 2689   class class class wbr 3936  cmpt 3996  1-1-ontowf1o 5129  cen 6639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-en 6642
This theorem is referenced by:  en2i  6671  map1  6713
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