ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1ocnvd GIF version

Theorem f1ocnvd 5702
Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
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
f1ocnvd (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)   𝑊(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem f1ocnvd
StepHypRef Expression
1 f1od.2 . . . . 5 ((𝜑𝑥𝐴) → 𝐶𝑊)
21ralrimiva 2392 . . . 4 (𝜑 → ∀𝑥𝐴 𝐶𝑊)
3 f1od.1 . . . . 5 𝐹 = (𝑥𝐴𝐶)
43fnmpt 5025 . . . 4 (∀𝑥𝐴 𝐶𝑊𝐹 Fn 𝐴)
52, 4syl 14 . . 3 (𝜑𝐹 Fn 𝐴)
6 f1od.3 . . . . . 6 ((𝜑𝑦𝐵) → 𝐷𝑋)
76ralrimiva 2392 . . . . 5 (𝜑 → ∀𝑦𝐵 𝐷𝑋)
8 eqid 2040 . . . . . 6 (𝑦𝐵𝐷) = (𝑦𝐵𝐷)
98fnmpt 5025 . . . . 5 (∀𝑦𝐵 𝐷𝑋 → (𝑦𝐵𝐷) Fn 𝐵)
107, 9syl 14 . . . 4 (𝜑 → (𝑦𝐵𝐷) Fn 𝐵)
11 f1od.4 . . . . . . 7 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
1211opabbidv 3823 . . . . . 6 (𝜑 → {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = 𝐷)})
13 df-mpt 3820 . . . . . . . . 9 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
143, 13eqtri 2060 . . . . . . . 8 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
1514cnveqi 4510 . . . . . . 7 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
16 cnvopab 4726 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
1715, 16eqtri 2060 . . . . . 6 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
18 df-mpt 3820 . . . . . 6 (𝑦𝐵𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = 𝐷)}
1912, 17, 183eqtr4g 2097 . . . . 5 (𝜑𝐹 = (𝑦𝐵𝐷))
2019fneq1d 4989 . . . 4 (𝜑 → (𝐹 Fn 𝐵 ↔ (𝑦𝐵𝐷) Fn 𝐵))
2110, 20mpbird 156 . . 3 (𝜑𝐹 Fn 𝐵)
22 dff1o4 5134 . . 3 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
235, 21, 22sylanbrc 394 . 2 (𝜑𝐹:𝐴1-1-onto𝐵)
2423, 19jca 290 1 (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  wral 2306  {copab 3817  cmpt 3818  ccnv 4344   Fn wfn 4897  1-1-ontowf1o 4901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909
This theorem is referenced by:  f1od  5703  f1ocnv2d  5704
  Copyright terms: Public domain W3C validator