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Theorem f1ocnv2d 7687
Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
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
f1ocnv2d (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
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 f1ocnv2d
StepHypRef Expression
1 f1od.1 . 2 𝐹 = (𝑥𝐴𝐶)
2 f1o2d.2 . 2 ((𝜑𝑥𝐴) → 𝐶𝐵)
3 f1o2d.3 . 2 ((𝜑𝑦𝐵) → 𝐷𝐴)
4 eleq1a 2835 . . . . . 6 (𝐶𝐵 → (𝑦 = 𝐶𝑦𝐵))
52, 4syl 17 . . . . 5 ((𝜑𝑥𝐴) → (𝑦 = 𝐶𝑦𝐵))
65impr 454 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦 = 𝐶)) → 𝑦𝐵)
7 f1o2d.4 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))
87biimpar 477 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑦 = 𝐶) → 𝑥 = 𝐷)
98exp42 435 . . . . . 6 (𝜑 → (𝑥𝐴 → (𝑦𝐵 → (𝑦 = 𝐶𝑥 = 𝐷))))
109com34 91 . . . . 5 (𝜑 → (𝑥𝐴 → (𝑦 = 𝐶 → (𝑦𝐵𝑥 = 𝐷))))
1110imp32 418 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦 = 𝐶)) → (𝑦𝐵𝑥 = 𝐷))
126, 11jcai 516 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦 = 𝐶)) → (𝑦𝐵𝑥 = 𝐷))
13 eleq1a 2835 . . . . . 6 (𝐷𝐴 → (𝑥 = 𝐷𝑥𝐴))
143, 13syl 17 . . . . 5 ((𝜑𝑦𝐵) → (𝑥 = 𝐷𝑥𝐴))
1514impr 454 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑥 = 𝐷)) → 𝑥𝐴)
167biimpa 476 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑥 = 𝐷) → 𝑦 = 𝐶)
1716exp42 435 . . . . . . 7 (𝜑 → (𝑥𝐴 → (𝑦𝐵 → (𝑥 = 𝐷𝑦 = 𝐶))))
1817com23 86 . . . . . 6 (𝜑 → (𝑦𝐵 → (𝑥𝐴 → (𝑥 = 𝐷𝑦 = 𝐶))))
1918com34 91 . . . . 5 (𝜑 → (𝑦𝐵 → (𝑥 = 𝐷 → (𝑥𝐴𝑦 = 𝐶))))
2019imp32 418 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑥 = 𝐷)) → (𝑥𝐴𝑦 = 𝐶))
2115, 20jcai 516 . . 3 ((𝜑 ∧ (𝑦𝐵𝑥 = 𝐷)) → (𝑥𝐴𝑦 = 𝐶))
2212, 21impbida 800 . 2 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
231, 2, 3, 22f1ocnvd 7685 1 (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  cmpt 5224  ccnv 5683  1-1-ontowf1o 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567
This theorem is referenced by:  f1o2d  7688  negiso  12249  iccf1o  13537  bitsf1ocnv  16482  grpinvcnv  19025  grplactcnv  19062  issrngd  20857  opncldf1  23093  txhmeo  23812  ptuncnv  23816  icopnfcnv  24974  iccpnfcnv  24976  gsumwrd2dccatlem  33070  xrge0iifcnv  33933  rfovcnvf1od  44022
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