Theorem f1oresf1o2 43539

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 6615	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
4	1, 3	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
5	4	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐹:𝐴⟶𝐵)
6	2	sselda 3967	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐴)
7	5, 6	jca 514	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴))
8	7	3adant3 1128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → (𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴))
9		ffvelrn 6849	. . . . . . 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 2900	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
12	11	3ad2ant3 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
13	10, 12	mpbid 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐵)
14		eqcom 2828	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
15		f1oresf1o2.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑥 ∈ 𝐷 ↔ 𝜒))
16	15	biimpd 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑥 ∈ 𝐷 → 𝜒))
17	16	ex 415	. . . . . . . 8 ⊢ (𝜑 → (𝑦 = (𝐹‘𝑥) → (𝑥 ∈ 𝐷 → 𝜒)))
18	14, 17	syl5bi 244	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑥) = 𝑦 → (𝑥 ∈ 𝐷 → 𝜒)))
19	18	com23 86	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐷 → ((𝐹‘𝑥) = 𝑦 → 𝜒)))
20	19	3imp 1107	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → 𝜒)
21	13, 20	jca 514	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → (𝑦 ∈ 𝐵 ∧ 𝜒))
22	21	rexlimdv3a 3286	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦 → (𝑦 ∈ 𝐵 ∧ 𝜒)))
23		f1ofo 6622	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
24	1, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)
25		foelrni 6727	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
26	24, 25	sylan 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
27	26	ex 415	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦))
28		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑥𝜑
29		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑥𝜒
30		nfre1 3306	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦
31	29, 30	nfim 1897	. . . . . 6 ⊢ Ⅎ𝑥(𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)
32		rspe 3304	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) = 𝑦) → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)
33	32	expcom 416	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝑥 ∈ 𝐷 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))
34	33	eqcoms 2829	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥 ∈ 𝐷 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))
35	34	adantl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑥 ∈ 𝐷 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))
36	15, 35	sylbird 262	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))
37	36	ex 415	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 = (𝐹‘𝑥) → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)))
38	37	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)))
39	14, 38	syl5bi 244	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦)))
40	39	ex 415	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 → (𝜒 → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))))
41	28, 31, 40	rexlimd 3317	. . . . 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 413	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜒) → ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦))
44	22, 43	impbid 214	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝜒)))
45	1, 2, 44	f1oresf1o 43538	1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∃wrex 3139  {crab 3142   ⊆ wss 3936   ↾ cres 5557  ⟶wf 6351  –onto→wfo 6353  –1-1-onto→wf1o 6354  ‘cfv 6355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363

This theorem is referenced by: (None)