Theorem f1oresf1o 47994

Description: Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.)

Hypotheses

Ref	Expression
f1oresf1o.1	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
f1oresf1o.2	⊢ (𝜑 → 𝐷 ⊆ 𝐴)
f1oresf1o.3	⊢ (𝜑 → (∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝜒)))

Assertion

Ref	Expression
f1oresf1o	⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

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

Allowed substitution hints:   𝜒(𝑦)   𝐴(𝑦)   𝐵(𝑦)

Proof of Theorem f1oresf1o

Step	Hyp	Ref	Expression
1		f1oresf1o.1	. . . 4 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
2		f1of1 6819	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
4		f1oresf1o.2	. . 3 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
5		f1ores 6835	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷))
6	3, 4, 5	syl2anc 595	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷))
7		f1ofun 6822	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹)
8	1, 7	syl 18	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
9		f1odm 6824	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴)
10	1, 9	syl 18	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
11	4, 10	sseqtrrd 3973	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹)
12		dfimafn 6943	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐷 ⊆ dom 𝐹) → (𝐹 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦})
13	8, 11, 12	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦})
14		f1oresf1o.3	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝜒)))
15	14	abbidv 2827	. . . . 5 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)})
16		df-rab 3415	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜒} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)}
17	15, 16	eqtr4di 2814	. . . 4 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦} = {𝑦 ∈ 𝐵 ∣ 𝜒})
18	13, 17	eqtr2d 2797	. . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} = (𝐹 “ 𝐷))
19	18	f1oeq3d 6817	. 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒} ↔ (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷)))
20	6, 19	mpbird 260	1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  {cab 2739  ∃wrex 3087  {crab 3414   ⊆ wss 3904  dom cdm 5661   ↾ cres 5663   “ cima 5664  Fun wfun 6530  –1-1→wf1 6533  –1-1-onto→wf1o 6535  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by:  f1oresf1o2  47995