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Theorem f1ocnvd 5940
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 2482 . . . 4 (𝜑 → ∀𝑥𝐴 𝐶𝑊)
3 f1od.1 . . . . 5 𝐹 = (𝑥𝐴𝐶)
43fnmpt 5219 . . . 4 (∀𝑥𝐴 𝐶𝑊𝐹 Fn 𝐴)
52, 4syl 14 . . 3 (𝜑𝐹 Fn 𝐴)
6 f1od.3 . . . . . 6 ((𝜑𝑦𝐵) → 𝐷𝑋)
76ralrimiva 2482 . . . . 5 (𝜑 → ∀𝑦𝐵 𝐷𝑋)
8 eqid 2117 . . . . . 6 (𝑦𝐵𝐷) = (𝑦𝐵𝐷)
98fnmpt 5219 . . . . 5 (∀𝑦𝐵 𝐷𝑋 → (𝑦𝐵𝐷) Fn 𝐵)
107, 9syl 14 . . . 4 (𝜑 → (𝑦𝐵𝐷) Fn 𝐵)
11 f1od.4 . . . . . . 7 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
1211opabbidv 3964 . . . . . 6 (𝜑 → {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = 𝐷)})
13 df-mpt 3961 . . . . . . . . 9 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
143, 13eqtri 2138 . . . . . . . 8 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
1514cnveqi 4684 . . . . . . 7 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
16 cnvopab 4910 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
1715, 16eqtri 2138 . . . . . 6 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
18 df-mpt 3961 . . . . . 6 (𝑦𝐵𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = 𝐷)}
1912, 17, 183eqtr4g 2175 . . . . 5 (𝜑𝐹 = (𝑦𝐵𝐷))
2019fneq1d 5183 . . . 4 (𝜑 → (𝐹 Fn 𝐵 ↔ (𝑦𝐵𝐷) Fn 𝐵))
2110, 20mpbird 166 . . 3 (𝜑𝐹 Fn 𝐵)
22 dff1o4 5343 . . 3 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
235, 21, 22sylanbrc 413 . 2 (𝜑𝐹:𝐴1-1-onto𝐵)
2423, 19jca 304 1 (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465  wral 2393  {copab 3958  cmpt 3959  ccnv 4508   Fn wfn 5088  1-1-ontowf1o 5092
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100
This theorem is referenced by:  f1od  5941  f1ocnv2d  5942
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