Theorem f1oresf1o 46296

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 6831	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
4		f1oresf1o.2	. . 3 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
5		f1ores 6846	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷))
6	3, 4, 5	syl2anc 582	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷))
7		f1ofun 6834	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹)
8	1, 7	syl 17	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
9		f1odm 6836	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴)
10	1, 9	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
11	4, 10	sseqtrrd 4022	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹)
12		dfimafn 6953	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐷 ⊆ dom 𝐹) → (𝐹 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦})
13	8, 11, 12	syl2anc 582	. . . 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 2799	. . . . 5 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)})
16		df-rab 3431	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜒} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)}
17	15, 16	eqtr4di 2788	. . . 4 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦} = {𝑦 ∈ 𝐵 ∣ 𝜒})
18	13, 17	eqtr2d 2771	. . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} = (𝐹 “ 𝐷))
19	18	f1oeq3d 6829	. 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒} ↔ (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷)))
20	6, 19	mpbird 256	1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {cab 2707  ∃wrex 3068  {crab 3430   ⊆ wss 3947  dom cdm 5675   ↾ cres 5677   “ cima 5678  Fun wfun 6536  –1-1→wf1 6539  –1-1-onto→wf1o 6541  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550

This theorem is referenced by:  f1oresf1o2  46297