Theorem f1oresf1orab 44668

Description: Build a bijection by restricting the domain of a bijection. (Contributed by AV, 1-Aug-2022.)

Hypotheses

Ref	Expression
f1oresf1orab.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
f1oresf1orab.2	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
f1oresf1orab.3	⊢ (𝜑 → 𝐷 ⊆ 𝐴)
f1oresf1orab.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → (𝜒 ↔ 𝑥 ∈ 𝐷))

Assertion

Ref	Expression
f1oresf1orab	⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷,𝑦   𝜑,𝑥,𝑦   𝜒,𝑥

Allowed substitution hints:   𝜒(𝑦)   𝐶(𝑥)   𝐹(𝑥,𝑦)

Proof of Theorem f1oresf1orab

Step	Hyp	Ref	Expression
1		f1oresf1orab.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
2		f1oresf1orab.2	. . 3 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
3		f1oresf1orab.4	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → (𝜒 ↔ 𝑥 ∈ 𝐷))
4	1, 2, 3	f1oresrab 6981	. 2 ⊢ (𝜑 → (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷}):{𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})
5		f1oresf1orab.3	. . . . . 6 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
6		dfss7 4171	. . . . . 6 ⊢ (𝐷 ⊆ 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷} = 𝐷)
7	5, 6	sylib 217	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷} = 𝐷)
8	7	eqcomd 2744	. . . 4 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷})
9	8	reseq2d 5880	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷}))
10		eqidd 2739	. . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} = {𝑦 ∈ 𝐵 ∣ 𝜒})
11	9, 8, 10	f1oeq123d 6694	. 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒} ↔ (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷}):{𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒}))
12	4, 11	mpbird 256	1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {crab 3067   ⊆ wss 3883   ↦ cmpt 5153   ↾ cres 5582  –1-1-onto→wf1o 6417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425

This theorem is referenced by: (None)