Theorem f1oresf1o2 48576

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⟶1-1onto𝐵)
f1oresf1o2.2	⊢ (𝜑 → 𝐷 ⊆ 𝐴)
f1oresf1o2.3	⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑥 ∈ 𝐷 ↔ 𝜒))

Assertion

Ref	Expression
f1oresf1o2	⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷⟶1-1onto{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⟶1-1onto𝐵)
2		f1oresf1o2.2	. 2 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
3		f1of 6849	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → 𝐹:𝐴⟶𝐵)
4	1, 3	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
5	4	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐹:𝐴⟶𝐵)
6	2	sselda 3913	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐴)
7	5, 6	jca 521	. . . . . . . 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 7108	. . . . . . 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 2852	. . . . . . 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 418	. . . . . . . 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 521	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → (𝑦 ∈ 𝐵 ∧ 𝜒))
22	21	rexlimdv3a 3171	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦 → (𝑦 ∈ 𝐵 ∧ 𝜒)))
23		f1ofo 6857	. . . . . . . 8 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → 𝐹:𝐴⟶onto𝐵)
24	1, 23	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶onto𝐵)
25		foelcdmi 6972	. . . . . . 7 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
26	24, 25	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
27	26	ex 418	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦))
28		nfv 1949	. . . . . 6 ⊢ Ⅎ𝑥𝜑
29		nfv 1949	. . . . . . 7 ⊢ Ⅎ𝑥𝜒
30		nfre1 3291	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦
31	29, 30	nfim 1931	. . . . . 6 ⊢ Ⅎ𝑥(𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)
32		rspe 3256	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)
33	32	expcom 419	. . . . . . . . . . . . 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 487	. . . . . . . . . . 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 418	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 = (𝐹‘𝑥) → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)))
38	37	adantr 486	. . . . . . . 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 418	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))))
41	28, 31, 40	rexlimd 3273	. . . . 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 416	. . 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 48575	1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷⟶1-1onto{𝑦 ∈ 𝐵 ∣ 𝜒})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {crab 3416   ⊆ wss 3900   ↾ cres 5658  ⟶wf 6557  ⟶ontowfo 6559  ⟶1-1ontowf1o 6560  ‘cfv 6561

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569

This theorem is used by: (None)