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Theorem usgrexilem 27209
Description: Lemma for usgrexi 27210. (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 6625 . . . 4 ( I ↾ 𝑃):𝑃1-1-onto𝑃
2 f1of1 6587 . . . 4 (( I ↾ 𝑃):𝑃1-1-onto𝑃 → ( I ↾ 𝑃):𝑃1-1𝑃)
31, 2ax-mp 5 . . 3 ( I ↾ 𝑃):𝑃1-1𝑃
4 dmresi 5894 . . . 4 dom ( I ↾ 𝑃) = 𝑃
5 f1eq2 6544 . . . 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 234 . 2 ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1𝑃
8 usgrexi.p . . . 4 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
98eqcomi 2830 . . 3 {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = 𝑃
10 f1eq3 6545 . . 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 261 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 209   = wceq 1538  wcel 2115  {crab 3130  𝒫 cpw 4512   I cid 5432  dom cdm 5528  cres 5530  1-1wf1 6325  1-1-ontowf1o 6327  cfv 6328  2c2 11670  chash 13674
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 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
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 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335
This theorem is referenced by:  usgrexi  27210  structtousgr  27214
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