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Theorem f1od 5731
Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
Hypotheses
Ref Expression
f1od.1 𝐹 = (𝑥𝐴𝐶)
f1od.2 ((𝜑𝑥𝐴) → 𝐶𝑊)
f1od.3 ((𝜑𝑦𝐵) → 𝐷𝑋)
f1od.4 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
Assertion
Ref Expression
f1od (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)   𝑊(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem f1od
StepHypRef Expression
1 f1od.1 . . 3 𝐹 = (𝑥𝐴𝐶)
2 f1od.2 . . 3 ((𝜑𝑥𝐴) → 𝐶𝑊)
3 f1od.3 . . 3 ((𝜑𝑦𝐵) → 𝐷𝑋)
4 f1od.4 . . 3 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
51, 2, 3, 4f1ocnvd 5730 . 2 (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
65simpld 109 1 (𝜑𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  cmpt 3846  ccnv 4372  1-1-ontowf1o 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937
This theorem is referenced by:  cnvf1o  5874  en2d  6279
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