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Theorem f1ocnvd 5644
Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
f1od.1 𝐹 = (x A𝐶)
f1od.2 ((φ x A) → 𝐶 𝑊)
f1od.3 ((φ y B) → 𝐷 𝑋)
f1od.4 (φ → ((x A y = 𝐶) ↔ (y B x = 𝐷)))
Assertion
Ref Expression
f1ocnvd (φ → (𝐹:A1-1-ontoB 𝐹 = (y B𝐷)))
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 ((φ x A) → 𝐶 𝑊)
21ralrimiva 2386 . . . 4 (φx A 𝐶 𝑊)
3 f1od.1 . . . . 5 𝐹 = (x A𝐶)
43fnmpt 4968 . . . 4 (x A 𝐶 𝑊𝐹 Fn A)
52, 4syl 14 . . 3 (φ𝐹 Fn A)
6 f1od.3 . . . . . 6 ((φ y B) → 𝐷 𝑋)
76ralrimiva 2386 . . . . 5 (φy B 𝐷 𝑋)
8 eqid 2037 . . . . . 6 (y B𝐷) = (y B𝐷)
98fnmpt 4968 . . . . 5 (y B 𝐷 𝑋 → (y B𝐷) Fn B)
107, 9syl 14 . . . 4 (φ → (y B𝐷) Fn B)
11 f1od.4 . . . . . . 7 (φ → ((x A y = 𝐶) ↔ (y B x = 𝐷)))
1211opabbidv 3814 . . . . . 6 (φ → {⟨y, x⟩ ∣ (x A y = 𝐶)} = {⟨y, x⟩ ∣ (y B x = 𝐷)})
13 df-mpt 3811 . . . . . . . . 9 (x A𝐶) = {⟨x, y⟩ ∣ (x A y = 𝐶)}
143, 13eqtri 2057 . . . . . . . 8 𝐹 = {⟨x, y⟩ ∣ (x A y = 𝐶)}
1514cnveqi 4453 . . . . . . 7 𝐹 = {⟨x, y⟩ ∣ (x A y = 𝐶)}
16 cnvopab 4669 . . . . . . 7 {⟨x, y⟩ ∣ (x A y = 𝐶)} = {⟨y, x⟩ ∣ (x A y = 𝐶)}
1715, 16eqtri 2057 . . . . . 6 𝐹 = {⟨y, x⟩ ∣ (x A y = 𝐶)}
18 df-mpt 3811 . . . . . 6 (y B𝐷) = {⟨y, x⟩ ∣ (y B x = 𝐷)}
1912, 17, 183eqtr4g 2094 . . . . 5 (φ𝐹 = (y B𝐷))
2019fneq1d 4932 . . . 4 (φ → (𝐹 Fn B ↔ (y B𝐷) Fn B))
2110, 20mpbird 156 . . 3 (φ𝐹 Fn B)
22 dff1o4 5077 . . 3 (𝐹:A1-1-ontoB ↔ (𝐹 Fn A 𝐹 Fn B))
235, 21, 22sylanbrc 394 . 2 (φ𝐹:A1-1-ontoB)
2423, 19jca 290 1 (φ → (𝐹:A1-1-ontoB 𝐹 = (y B𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  {copab 3808  cmpt 3809  ccnv 4287   Fn wfn 4840  1-1-ontowf1o 4844
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by:  f1od  5645  f1ocnv2d  5646
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