Theorem usgrexilem 29580

Description: Lemma for usgrexi 29581. (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 6834	. . . 4 ⊢ ( I ↾ 𝑃):𝑃–1-1-onto→𝑃
2		f1of1 6794	. . . 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 6031	. . . 4 ⊢ dom ( I ↾ 𝑃) = 𝑃
5		f1eq2 6745	. . . 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 233	. 2 ⊢ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃
8		usgrexi.p	. . . 4 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
9	8	eqcomi 2765	. . 3 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = 𝑃
10		f1eq3 6746	. . 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 260	1 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1554   ∈ wcel 2136  {crab 3408  𝒫 cpw 4549   I cid 5534  dom cdm 5640   ↾ cres 5642  –1-1→wf1 6507  –1-1-onto→wf1o 6509  ‘cfv 6510  2c2 12262  ♯chash 14333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-br 5095  df-opab 5157  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517

This theorem is referenced by:  usgrexi  29581  structtousgr  29585