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Theorem f1ocnv2d 5974
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 2211 . . . . . 6 (𝐶𝐵 → (𝑦 = 𝐶𝑦𝐵))
52, 4syl 14 . . . . 5 ((𝜑𝑥𝐴) → (𝑦 = 𝐶𝑦𝐵))
65impr 376 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦 = 𝐶)) → 𝑦𝐵)
7 f1o2d.4 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))
87biimpar 295 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑦 = 𝐶) → 𝑥 = 𝐷)
98exp42 368 . . . . . 6 (𝜑 → (𝑥𝐴 → (𝑦𝐵 → (𝑦 = 𝐶𝑥 = 𝐷))))
109com34 83 . . . . 5 (𝜑 → (𝑥𝐴 → (𝑦 = 𝐶 → (𝑦𝐵𝑥 = 𝐷))))
1110imp32 255 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦 = 𝐶)) → (𝑦𝐵𝑥 = 𝐷))
126, 11jcai 309 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦 = 𝐶)) → (𝑦𝐵𝑥 = 𝐷))
13 eleq1a 2211 . . . . . 6 (𝐷𝐴 → (𝑥 = 𝐷𝑥𝐴))
143, 13syl 14 . . . . 5 ((𝜑𝑦𝐵) → (𝑥 = 𝐷𝑥𝐴))
1514impr 376 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑥 = 𝐷)) → 𝑥𝐴)
167biimpa 294 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑥 = 𝐷) → 𝑦 = 𝐶)
1716exp42 368 . . . . . . 7 (𝜑 → (𝑥𝐴 → (𝑦𝐵 → (𝑥 = 𝐷𝑦 = 𝐶))))
1817com23 78 . . . . . 6 (𝜑 → (𝑦𝐵 → (𝑥𝐴 → (𝑥 = 𝐷𝑦 = 𝐶))))
1918com34 83 . . . . 5 (𝜑 → (𝑦𝐵 → (𝑥 = 𝐷 → (𝑥𝐴𝑦 = 𝐶))))
2019imp32 255 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑥 = 𝐷)) → (𝑥𝐴𝑦 = 𝐶))
2115, 20jcai 309 . . 3 ((𝜑 ∧ (𝑦𝐵𝑥 = 𝐷)) → (𝑥𝐴𝑦 = 𝐶))
2212, 21impbida 585 . 2 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
231, 2, 3, 22f1ocnvd 5972 1 (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  cmpt 3989  ccnv 4538  1-1-ontowf1o 5122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130
This theorem is referenced by:  f1o2d  5975  negf1o  8156  negiso  8725  iccf1o  9799  xrnegiso  11043  txhmeo  12502
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