Theorem f1oresf1orab 47320

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 7055	. 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 4196	. . . . . 6 ⊢ (𝐷 ⊆ 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷} = 𝐷)
7	5, 6	sylib 218	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷} = 𝐷)
8	7	eqcomd 2737	. . . 4 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷})
9	8	reseq2d 5923	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷}))
10		eqidd 2732	. . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} = {𝑦 ∈ 𝐵 ∣ 𝜒})
11	9, 8, 10	f1oeq123d 6752	. 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 257	1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  {crab 3395   ⊆ wss 3897   ↦ cmpt 5167   ↾ cres 5613  –1-1-onto→wf1o 6475

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

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-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483

This theorem is referenced by: (None)