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Theorem f1ocnvd 5727
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 2407 . . . 4 (𝜑 → ∀𝑥𝐴 𝐶𝑊)
3 f1od.1 . . . . 5 𝐹 = (𝑥𝐴𝐶)
43fnmpt 5050 . . . 4 (∀𝑥𝐴 𝐶𝑊𝐹 Fn 𝐴)
52, 4syl 14 . . 3 (𝜑𝐹 Fn 𝐴)
6 f1od.3 . . . . . 6 ((𝜑𝑦𝐵) → 𝐷𝑋)
76ralrimiva 2407 . . . . 5 (𝜑 → ∀𝑦𝐵 𝐷𝑋)
8 eqid 2054 . . . . . 6 (𝑦𝐵𝐷) = (𝑦𝐵𝐷)
98fnmpt 5050 . . . . 5 (∀𝑦𝐵 𝐷𝑋 → (𝑦𝐵𝐷) Fn 𝐵)
107, 9syl 14 . . . 4 (𝜑 → (𝑦𝐵𝐷) Fn 𝐵)
11 f1od.4 . . . . . . 7 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
1211opabbidv 3848 . . . . . 6 (𝜑 → {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = 𝐷)})
13 df-mpt 3845 . . . . . . . . 9 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
143, 13eqtri 2074 . . . . . . . 8 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
1514cnveqi 4535 . . . . . . 7 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
16 cnvopab 4751 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
1715, 16eqtri 2074 . . . . . 6 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
18 df-mpt 3845 . . . . . 6 (𝑦𝐵𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = 𝐷)}
1912, 17, 183eqtr4g 2111 . . . . 5 (𝜑𝐹 = (𝑦𝐵𝐷))
2019fneq1d 5014 . . . 4 (𝜑 → (𝐹 Fn 𝐵 ↔ (𝑦𝐵𝐷) Fn 𝐵))
2110, 20mpbird 160 . . 3 (𝜑𝐹 Fn 𝐵)
22 dff1o4 5159 . . 3 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
235, 21, 22sylanbrc 402 . 2 (𝜑𝐹:𝐴1-1-onto𝐵)
2423, 19jca 294 1 (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1257  wcel 1407  wral 2321  {copab 3842  cmpt 3843  ccnv 4369   Fn wfn 4922  1-1-ontowf1o 4926
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 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790  df-opab 3844  df-mpt 3845  df-id 4055  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934
This theorem is referenced by:  f1od  5728  f1ocnv2d  5729
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