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Theorem en2d 6439
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 2085 . . 3 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
4 en2d.3 . . . 4 (𝜑 → (𝑥𝐴𝐶 ∈ V))
54imp 122 . . 3 ((𝜑𝑥𝐴) → 𝐶 ∈ V)
6 en2d.4 . . . 4 (𝜑 → (𝑦𝐵𝐷 ∈ V))
76imp 122 . . 3 ((𝜑𝑦𝐵) → 𝐷 ∈ V)
8 en2d.5 . . 3 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
93, 5, 7, 8f1od 5806 . 2 (𝜑 → (𝑥𝐴𝐶):𝐴1-1-onto𝐵)
10 f1oen2g 6426 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ (𝑥𝐴𝐶):𝐴1-1-onto𝐵) → 𝐴𝐵)
111, 2, 9, 10syl3anc 1172 1 (𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wcel 1436  Vcvv 2615   class class class wbr 3822  cmpt 3876  1-1-ontowf1o 4982  cen 6409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-mpt 3878  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-en 6412
This theorem is referenced by:  en2i  6441  map1  6483
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