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Theorem en2d 8533
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 2818 . . 3 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
4 en2d.3 . . . 4 (𝜑 → (𝑥𝐴𝐶 ∈ V))
54imp 407 . . 3 ((𝜑𝑥𝐴) → 𝐶 ∈ V)
6 en2d.4 . . . 4 (𝜑 → (𝑦𝐵𝐷 ∈ V))
76imp 407 . . 3 ((𝜑𝑦𝐵) → 𝐷 ∈ V)
8 en2d.5 . . 3 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
93, 5, 7, 8f1od 7386 . 2 (𝜑 → (𝑥𝐴𝐶):𝐴1-1-onto𝐵)
10 f1oen2g 8514 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ (𝑥𝐴𝐶):𝐴1-1-onto𝐵) → 𝐴𝐵)
111, 2, 9, 10syl3anc 1363 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  Vcvv 3492   class class class wbr 5057  cmpt 5137  1-1-ontowf1o 6347  cen 8494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-en 8498
This theorem is referenced by:  en2i  8535  mapsnend  8576  snmapen  8578  gicsubgen  18356  lzenom  39245
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