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Theorem f1ocnv2d 7668
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 2823 . . . . . 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 2823 . . . . . 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 7666 1 (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  cmpt 5225  ccnv 5671  1-1-ontowf1o 6541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549
This theorem is referenced by:  f1o2d  7669  negiso  12216  iccf1o  13497  bitsf1ocnv  16410  grpinvcnv  18954  grplactcnv  18990  issrngd  20730  opncldf1  22975  txhmeo  23694  ptuncnv  23698  icopnfcnv  24854  iccpnfcnv  24856  xrge0iifcnv  33470  rfovcnvf1od  43357
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