Theorem f1oresf1orab 47844

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 7104	. 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 4201	. . . . . 6 ⊢ (𝐷 ⊆ 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷} = 𝐷)
7	5, 6	sylib 220	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷} = 𝐷)
8	7	eqcomd 2767	. . . 4 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷})
9	8	reseq2d 5961	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷}))
10		eqidd 2762	. . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} = {𝑦 ∈ 𝐵 ∣ 𝜒})
11	9, 8, 10	f1oeq123d 6795	. 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 259	1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  {crab 3413   ⊆ wss 3902   ↦ cmpt 5178   ↾ cres 5645  –1-1-onto→wf1o 6515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523

This theorem is referenced by: (None)