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Theorem f1oresf1o 43772
Description: Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.)
Hypotheses
Ref Expression
f1oresf1o.1 (𝜑𝐹:𝐴1-1-onto𝐵)
f1oresf1o.2 (𝜑𝐷𝐴)
f1oresf1o.3 (𝜑 → (∃𝑥𝐷 (𝐹𝑥) = 𝑦 ↔ (𝑦𝐵𝜒)))
Assertion
Ref Expression
f1oresf1o (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑦,𝐷   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦   𝜒,𝑥
Allowed substitution hints:   𝜒(𝑦)   𝐴(𝑦)   𝐵(𝑦)

Proof of Theorem f1oresf1o
StepHypRef Expression
1 f1oresf1o.1 . . . 4 (𝜑𝐹:𝐴1-1-onto𝐵)
2 f1of1 6605 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
31, 2syl 17 . . 3 (𝜑𝐹:𝐴1-1𝐵)
4 f1oresf1o.2 . . 3 (𝜑𝐷𝐴)
5 f1ores 6620 . . 3 ((𝐹:𝐴1-1𝐵𝐷𝐴) → (𝐹𝐷):𝐷1-1-onto→(𝐹𝐷))
63, 4, 5syl2anc 587 . 2 (𝜑 → (𝐹𝐷):𝐷1-1-onto→(𝐹𝐷))
7 f1ofun 6608 . . . . . 6 (𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
81, 7syl 17 . . . . 5 (𝜑 → Fun 𝐹)
9 f1odm 6610 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵 → dom 𝐹 = 𝐴)
101, 9syl 17 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
114, 10sseqtrrd 3994 . . . . 5 (𝜑𝐷 ⊆ dom 𝐹)
12 dfimafn 6719 . . . . 5 ((Fun 𝐹𝐷 ⊆ dom 𝐹) → (𝐹𝐷) = {𝑦 ∣ ∃𝑥𝐷 (𝐹𝑥) = 𝑦})
138, 11, 12syl2anc 587 . . . 4 (𝜑 → (𝐹𝐷) = {𝑦 ∣ ∃𝑥𝐷 (𝐹𝑥) = 𝑦})
14 f1oresf1o.3 . . . . . 6 (𝜑 → (∃𝑥𝐷 (𝐹𝑥) = 𝑦 ↔ (𝑦𝐵𝜒)))
1514abbidv 2888 . . . . 5 (𝜑 → {𝑦 ∣ ∃𝑥𝐷 (𝐹𝑥) = 𝑦} = {𝑦 ∣ (𝑦𝐵𝜒)})
16 df-rab 3142 . . . . 5 {𝑦𝐵𝜒} = {𝑦 ∣ (𝑦𝐵𝜒)}
1715, 16syl6eqr 2877 . . . 4 (𝜑 → {𝑦 ∣ ∃𝑥𝐷 (𝐹𝑥) = 𝑦} = {𝑦𝐵𝜒})
1813, 17eqtr2d 2860 . . 3 (𝜑 → {𝑦𝐵𝜒} = (𝐹𝐷))
1918f1oeq3d 6603 . 2 (𝜑 → ((𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒} ↔ (𝐹𝐷):𝐷1-1-onto→(𝐹𝐷)))
206, 19mpbird 260 1 (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {cab 2802  wrex 3134  {crab 3137  wss 3919  dom cdm 5542  cres 5544  cima 5545  Fun wfun 6337  1-1wf1 6340  1-1-ontowf1o 6342  cfv 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351
This theorem is referenced by:  f1oresf1o2  43773
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