Theorem usgrexilem 29411

Description: Lemma for usgrexi 29412. (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 6797	. . . 4 ⊢ ( I ↾ 𝑃):𝑃–1-1-onto→𝑃
2		f1of1 6758	. . . 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 5998	. . . 4 ⊢ dom ( I ↾ 𝑃) = 𝑃
5		f1eq2 6711	. . . 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 2739	. . 3 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = 𝑃
10		f1eq3 6712	. . 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 1541   ∈ wcel 2110  {crab 3393  𝒫 cpw 4548   I cid 5508  dom cdm 5614   ↾ cres 5616  –1-1→wf1 6474  –1-1-onto→wf1o 6476  ‘cfv 6477  2c2 12172  ♯chash 14229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484

This theorem is referenced by:  usgrexi  29412  structtousgr  29416