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Theorem usgrexilem 26558
Description: Lemma for usgrexi 26559. (Contributed by AV, 12-Jan-2018.) (Revised by AV, 10-Nov-2021.)
Hypothesis
Ref Expression
usgrexi.p 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
Assertion
Ref Expression
usgrexilem (𝑉𝑊 → ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
Distinct variable group:   𝑥,𝑉
Allowed substitution hints:   𝑃(𝑥)   𝑊(𝑥)

Proof of Theorem usgrexilem
StepHypRef Expression
1 f1oi 6384 . . . 4 ( I ↾ 𝑃):𝑃1-1-onto𝑃
2 f1of1 6346 . . . 4 (( I ↾ 𝑃):𝑃1-1-onto𝑃 → ( I ↾ 𝑃):𝑃1-1𝑃)
31, 2ax-mp 5 . . 3 ( I ↾ 𝑃):𝑃1-1𝑃
4 dmresi 5663 . . . 4 dom ( I ↾ 𝑃) = 𝑃
5 f1eq2 6306 . . . 4 (dom ( I ↾ 𝑃) = 𝑃 → (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1𝑃 ↔ ( I ↾ 𝑃):𝑃1-1𝑃))
64, 5ax-mp 5 . . 3 (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1𝑃 ↔ ( I ↾ 𝑃):𝑃1-1𝑃)
73, 6mpbir 222 . 2 ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1𝑃
8 usgrexi.p . . . 4 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
98eqcomi 2811 . . 3 {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = 𝑃
10 f1eq3 6307 . . 3 ({𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = 𝑃 → (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1𝑃))
119, 10mp1i 13 . 2 (𝑉𝑊 → (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1𝑃))
127, 11mpbiri 249 1 (𝑉𝑊 → ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1637  wcel 2155  {crab 3096  𝒫 cpw 4345   I cid 5212  dom cdm 5305  cres 5307  1-1wf1 6092  1-1-ontowf1o 6094  cfv 6095  2c2 11350  chash 13331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pr 5090
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ral 3097  df-rex 3098  df-rab 3101  df-v 3389  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-sn 4365  df-pr 4367  df-op 4371  df-br 4838  df-opab 4900  df-id 5213  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102
This theorem is referenced by:  usgrexi  26559  structtousgr  26563
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