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Theorem en2d 6767
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 2177 . . 3 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
4 en2d.3 . . . 4 (𝜑 → (𝑥𝐴𝐶 ∈ V))
54imp 124 . . 3 ((𝜑𝑥𝐴) → 𝐶 ∈ V)
6 en2d.4 . . . 4 (𝜑 → (𝑦𝐵𝐷 ∈ V))
76imp 124 . . 3 ((𝜑𝑦𝐵) → 𝐷 ∈ V)
8 en2d.5 . . 3 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
93, 5, 7, 8f1od 6073 . 2 (𝜑 → (𝑥𝐴𝐶):𝐴1-1-onto𝐵)
10 f1oen2g 6754 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ (𝑥𝐴𝐶):𝐴1-1-onto𝐵) → 𝐴𝐵)
111, 2, 9, 10syl3anc 1238 1 (𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  Vcvv 2737   class class class wbr 4003  cmpt 4064  1-1-ontowf1o 5215  cen 6737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-en 6740
This theorem is referenced by:  en2i  6769  map1  6811
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