Theorem f1oresf1o2 47206

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 6862	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
4	1, 3	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
5	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐹:𝐴⟶𝐵)
6	2	sselda 4008	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐴)
7	5, 6	jca 511	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴))
8	7	3adant3 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → (𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴))
9		ffvelcdm 7115	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
10	8, 9	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) ∈ 𝐵)
11		eleq1 2832	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
12	11	3ad2ant3 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
13	10, 12	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐵)
14		eqcom 2747	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
15		f1oresf1o2.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑥 ∈ 𝐷 ↔ 𝜒))
16	15	biimpd 229	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑥 ∈ 𝐷 → 𝜒))
17	16	ex 412	. . . . . . . 8 ⊢ (𝜑 → (𝑦 = (𝐹‘𝑥) → (𝑥 ∈ 𝐷 → 𝜒)))
18	14, 17	biimtrid 242	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑥) = 𝑦 → (𝑥 ∈ 𝐷 → 𝜒)))
19	18	com23 86	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐷 → ((𝐹‘𝑥) = 𝑦 → 𝜒)))
20	19	3imp 1111	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → 𝜒)
21	13, 20	jca 511	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → (𝑦 ∈ 𝐵 ∧ 𝜒))
22	21	rexlimdv3a 3165	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦 → (𝑦 ∈ 𝐵 ∧ 𝜒)))
23		f1ofo 6869	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
24	1, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)
25		foelcdmi 6983	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
26	24, 25	sylan 579	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
27	26	ex 412	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦))
28		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑥𝜑
29		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑥𝜒
30		nfre1 3291	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦
31	29, 30	nfim 1895	. . . . . 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 413	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝑥 ∈ 𝐷 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))
34	33	eqcoms 2748	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥 ∈ 𝐷 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))
35	34	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑥 ∈ 𝐷 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))
36	15, 35	sylbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))
37	36	ex 412	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 = (𝐹‘𝑥) → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)))
38	37	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)))
39	14, 38	biimtrid 242	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)))
40	39	ex 412	. . . . . 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 47	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)))
43	42	impd 410	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜒) → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))
44	22, 43	impbid 212	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝜒)))
45	1, 2, 44	f1oresf1o 47205	1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∃wrex 3076  {crab 3443   ⊆ wss 3976   ↾ cres 5702  ⟶wf 6569  –onto→wfo 6571  –1-1-onto→wf1o 6572  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581

This theorem is referenced by: (None)