Theorem f1oresf1o2 48011

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

Hypotheses

Ref	Expression
f1oresf1o2.1	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
f1oresf1o2.2	⊢ (𝜑 → 𝐷 ⊆ 𝐴)
f1oresf1o2.3	⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑥 ∈ 𝐷 ↔ 𝜒))

Assertion

Ref	Expression
f1oresf1o2	⊢ (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 f1oresf1o2

Step	Hyp	Ref	Expression
1		f1oresf1o2.1	. 2 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
2		f1oresf1o2.2	. 2 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
3		f1of 6822	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
4	1, 3	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
5	4	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐹:𝐴⟶𝐵)
6	2	sselda 3938	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐴)
7	5, 6	jca 520	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴))
8	7	3adant3 1150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → (𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴))
9		ffvelcdm 7078	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
10	8, 9	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) ∈ 𝐵)
11		eleq1 2851	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
12	11	3ad2ant3 1153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
13	10, 12	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐵)
14		eqcom 2770	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
15		f1oresf1o2.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑥 ∈ 𝐷 ↔ 𝜒))
16	15	biimpd 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑥 ∈ 𝐷 → 𝜒))
17	16	ex 417	. . . . . . . 8 ⊢ (𝜑 → (𝑦 = (𝐹‘𝑥) → (𝑥 ∈ 𝐷 → 𝜒)))
18	14, 17	biimtrid 245	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑥) = 𝑦 → (𝑥 ∈ 𝐷 → 𝜒)))
19	18	com23 87	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐷 → ((𝐹‘𝑥) = 𝑦 → 𝜒)))
20	19	3imp 1128	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → 𝜒)
21	13, 20	jca 520	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → (𝑦 ∈ 𝐵 ∧ 𝜒))
22	21	rexlimdv3a 3170	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦 → (𝑦 ∈ 𝐵 ∧ 𝜒)))
23		f1ofo 6830	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
24	1, 23	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)
25		foelcdmi 6944	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
26	24, 25	sylan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
27	26	ex 417	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦))
28		nfv 1944	. . . . . 6 ⊢ Ⅎ𝑥𝜑
29		nfv 1944	. . . . . . 7 ⊢ Ⅎ𝑥𝜒
30		nfre1 3290	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦
31	29, 30	nfim 1926	. . . . . 6 ⊢ Ⅎ𝑥(𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)
32		rspe 3255	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)
33	32	expcom 418	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝑥 ∈ 𝐷 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))
34	33	eqcoms 2771	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥 ∈ 𝐷 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))
35	34	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑥 ∈ 𝐷 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))
36	15, 35	sylbird 263	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))
37	36	ex 417	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 = (𝐹‘𝑥) → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)))
38	37	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)))
39	14, 38	biimtrid 245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)))
40	39	ex 417	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))))
41	28, 31, 40	rexlimd 3272	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)))
42	27, 41	syld 48	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)))
43	42	impd 415	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜒) → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))
44	22, 43	impbid 215	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝜒)))
45	1, 2, 44	f1oresf1o 48010	1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  {crab 3416   ⊆ wss 3906   ↾ cres 5665  ⟶wf 6534  –onto→wfo 6536  –1-1-onto→wf1o 6537  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546

This theorem is referenced by: (None)