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Theorem en2d 8144
 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 2770 . . 3 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
4 en2d.3 . . . 4 (𝜑 → (𝑥𝐴𝐶 ∈ V))
54imp 393 . . 3 ((𝜑𝑥𝐴) → 𝐶 ∈ V)
6 en2d.4 . . . 4 (𝜑 → (𝑦𝐵𝐷 ∈ V))
76imp 393 . . 3 ((𝜑𝑦𝐵) → 𝐷 ∈ V)
8 en2d.5 . . 3 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
93, 5, 7, 8f1od 7031 . 2 (𝜑 → (𝑥𝐴𝐶):𝐴1-1-onto𝐵)
10 f1oen2g 8125 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ (𝑥𝐴𝐶):𝐴1-1-onto𝐵) → 𝐴𝐵)
111, 2, 9, 10syl3anc 1475 1 (𝜑𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1630   ∈ wcel 2144  Vcvv 3349   class class class wbr 4784   ↦ cmpt 4861  –1-1-onto→wf1o 6030   ≈ cen 8105 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-en 8109 This theorem is referenced by:  en2i  8146  mapsnend  8187  snmapen  8189  gicsubgen  17927  lzenom  37852
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