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Theorem f1oresf1orab 44481
Description: Build a bijection by restricting the domain of a bijection. (Contributed by AV, 1-Aug-2022.)
Hypotheses
Ref Expression
f1oresf1orab.1 𝐹 = (𝑥𝐴𝐶)
f1oresf1orab.2 (𝜑𝐹:𝐴1-1-onto𝐵)
f1oresf1orab.3 (𝜑𝐷𝐴)
f1oresf1orab.4 ((𝜑𝑥𝐴𝑦 = 𝐶) → (𝜒𝑥𝐷))
Assertion
Ref Expression
f1oresf1orab (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷,𝑦   𝜑,𝑥,𝑦   𝜒,𝑥
Allowed substitution hints:   𝜒(𝑦)   𝐶(𝑥)   𝐹(𝑥,𝑦)

Proof of Theorem f1oresf1orab
StepHypRef Expression
1 f1oresf1orab.1 . . 3 𝐹 = (𝑥𝐴𝐶)
2 f1oresf1orab.2 . . 3 (𝜑𝐹:𝐴1-1-onto𝐵)
3 f1oresf1orab.4 . . 3 ((𝜑𝑥𝐴𝑦 = 𝐶) → (𝜒𝑥𝐷))
41, 2, 3f1oresrab 6960 . 2 (𝜑 → (𝐹 ↾ {𝑥𝐴𝑥𝐷}):{𝑥𝐴𝑥𝐷}–1-1-onto→{𝑦𝐵𝜒})
5 f1oresf1orab.3 . . . . . 6 (𝜑𝐷𝐴)
6 dfss7 4169 . . . . . 6 (𝐷𝐴 ↔ {𝑥𝐴𝑥𝐷} = 𝐷)
75, 6sylib 221 . . . . 5 (𝜑 → {𝑥𝐴𝑥𝐷} = 𝐷)
87eqcomd 2744 . . . 4 (𝜑𝐷 = {𝑥𝐴𝑥𝐷})
98reseq2d 5865 . . 3 (𝜑 → (𝐹𝐷) = (𝐹 ↾ {𝑥𝐴𝑥𝐷}))
10 eqidd 2739 . . 3 (𝜑 → {𝑦𝐵𝜒} = {𝑦𝐵𝜒})
119, 8, 10f1oeq123d 6673 . 2 (𝜑 → ((𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒} ↔ (𝐹 ↾ {𝑥𝐴𝑥𝐷}):{𝑥𝐴𝑥𝐷}–1-1-onto→{𝑦𝐵𝜒}))
124, 11mpbird 260 1 (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1089   = wceq 1543  wcel 2111  {crab 3066  wss 3880  cmpt 5149  cres 5567  1-1-ontowf1o 6396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-br 5068  df-opab 5130  df-mpt 5150  df-id 5469  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-rn 5576  df-res 5577  df-ima 5578  df-fun 6399  df-fn 6400  df-f 6401  df-f1 6402  df-fo 6403  df-f1o 6404
This theorem is referenced by: (None)
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