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Theorem usgrexilem 26467
Description: Lemma for usgrexi 26468. (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 6287 . . . 4 ( I ↾ 𝑃):𝑃1-1-onto𝑃
2 f1of1 6249 . . . 4 (( I ↾ 𝑃):𝑃1-1-onto𝑃 → ( I ↾ 𝑃):𝑃1-1𝑃)
31, 2ax-mp 5 . . 3 ( I ↾ 𝑃):𝑃1-1𝑃
4 dmresi 5567 . . . 4 dom ( I ↾ 𝑃) = 𝑃
5 f1eq2 6210 . . . 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 221 . 2 ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1𝑃
8 usgrexi.p . . . 4 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
98eqcomi 2733 . . 3 {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = 𝑃
10 f1eq3 6211 . . 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 248 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 196   = wceq 1596  wcel 2103  {crab 3018  𝒫 cpw 4266   I cid 5127  dom cdm 5218  cres 5220  1-1wf1 5998  1-1-ontowf1o 6000  cfv 6001  2c2 11183  chash 13232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008
This theorem is referenced by:  usgrexi  26468  structtousgr  26472
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