MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1o2d Structured version   Visualization version   GIF version

Theorem f1o2d 6762
Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
Hypotheses
Ref Expression
f1od.1 𝐹 = (𝑥𝐴𝐶)
f1o2d.2 ((𝜑𝑥𝐴) → 𝐶𝐵)
f1o2d.3 ((𝜑𝑦𝐵) → 𝐷𝐴)
f1o2d.4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))
Assertion
Ref Expression
f1o2d (𝜑𝐹:𝐴1-1-onto𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem f1o2d
StepHypRef Expression
1 f1od.1 . . 3 𝐹 = (𝑥𝐴𝐶)
2 f1o2d.2 . . 3 ((𝜑𝑥𝐴) → 𝐶𝐵)
3 f1o2d.3 . . 3 ((𝜑𝑦𝐵) → 𝐷𝐴)
4 f1o2d.4 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))
51, 2, 3, 4f1ocnv2d 6761 . 2 (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
65simpld 473 1 (𝜑𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  cmpt 4637  ccnv 5027  1-1-ontowf1o 5789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797
This theorem is referenced by:  f1opw2  6763  en3d  7855  f1opwfi  8130  mapfien  8173  fin23lem22  9009  incexclem  14353  dvdsflip  14823  hashgcdlem  15277  grplmulf1o  17258  conjghm  17460  gapm  17508  psrbagconf1o  19141  hmeoimaf1o  21325  itg1mulc  23194  resinf1o  24003  eff1olem  24015  sqff1o  24625  dvdsppwf1o  24629  dvdsflf1o  24630  fcobij  28694
  Copyright terms: Public domain W3C validator