Theorem usgrexilem 29472

Description: Lemma for usgrexi 29473. (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	⊢ (𝑉 ∈ 𝑊 → ( 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

Step	Hyp	Ref	Expression
1		f1oi 6887	. . . 4 ⊢ ( I ↾ 𝑃):𝑃–1-1-onto→𝑃
2		f1of1 6848	. . . 4 ⊢ (( I ↾ 𝑃):𝑃–1-1-onto→𝑃 → ( I ↾ 𝑃):𝑃–1-1→𝑃)
3	1, 2	ax-mp 5	. . 3 ⊢ ( I ↾ 𝑃):𝑃–1-1→𝑃
4		dmresi 6072	. . . 4 ⊢ dom ( I ↾ 𝑃) = 𝑃
5		f1eq2 6801	. . . 4 ⊢ (dom ( I ↾ 𝑃) = 𝑃 → (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃 ↔ ( I ↾ 𝑃):𝑃–1-1→𝑃))
6	4, 5	ax-mp 5	. . 3 ⊢ (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃 ↔ ( I ↾ 𝑃):𝑃–1-1→𝑃)
7	3, 6	mpbir 231	. 2 ⊢ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃
8		usgrexi.p	. . . 4 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
9	8	eqcomi 2744	. . 3 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = 𝑃
10		f1eq3 6802	. . 3 ⊢ ({𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = 𝑃 → (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃))
11	9, 10	mp1i 13	. 2 ⊢ (𝑉 ∈ 𝑊 → (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃))
12	7, 11	mpbiri 258	1 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106  {crab 3433  𝒫 cpw 4605   I cid 5582  dom cdm 5689   ↾ cres 5691  –1-1→wf1 6560  –1-1-onto→wf1o 6562  ‘cfv 6563  2c2 12319  ♯chash 14366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570

This theorem is referenced by:  usgrexi  29473  structtousgr  29477