Theorem exfo 6377

Description: A relation equivalent to the existence of an onto mapping. The right-hand 𝑓 is not necessarily a function. (Contributed by NM, 20-Mar-2007.)

Assertion

Ref	Expression
exfo	⊢ (∃𝑓 𝑓:𝐴–onto→𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥))

Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝐵,𝑓,𝑥,𝑦

Proof of Theorem exfo

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffo4 6375	. . . 4 ⊢ (𝑓:𝐴–onto→𝐵 ↔ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥))
2		dff4 6373	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 ↔ (𝑓 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦))
3	2	simprbi 480	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦)
4	3	anim1i 592	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥) → (∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥))
5	1, 4	sylbi 207	. . 3 ⊢ (𝑓:𝐴–onto→𝐵 → (∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥))
6	5	eximi 1762	. 2 ⊢ (∃𝑓 𝑓:𝐴–onto→𝐵 → ∃𝑓(∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥))
7		brinxp 5181	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑓𝑦 ↔ 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
8	7	reubidva 3125	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ↔ ∃!𝑦 ∈ 𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
9	8	biimpd 219	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 → ∃!𝑦 ∈ 𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
10	9	ralimia 2950	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 → ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦)
11		inss2 3834	. . . . . . . . 9 ⊢ (𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
12	10, 11	jctil 560	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 → ((𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
13		dff4 6373	. . . . . . . 8 ⊢ ((𝑓 ∩ (𝐴 × 𝐵)):𝐴⟶𝐵 ↔ ((𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
14	12, 13	sylibr 224	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 → (𝑓 ∩ (𝐴 × 𝐵)):𝐴⟶𝐵)
15		rninxp 5573	. . . . . . . 8 ⊢ (ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥)
16	15	biimpri 218	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥 → ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵)
17	14, 16	anim12i 590	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥) → ((𝑓 ∩ (𝐴 × 𝐵)):𝐴⟶𝐵 ∧ ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵))
18		dffo2 6119	. . . . . 6 ⊢ ((𝑓 ∩ (𝐴 × 𝐵)):𝐴–onto→𝐵 ↔ ((𝑓 ∩ (𝐴 × 𝐵)):𝐴⟶𝐵 ∧ ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵))
19	17, 18	sylibr 224	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥) → (𝑓 ∩ (𝐴 × 𝐵)):𝐴–onto→𝐵)
20		vex 3203	. . . . . . 7 ⊢ 𝑓 ∈ V
21	20	inex1 4799	. . . . . 6 ⊢ (𝑓 ∩ (𝐴 × 𝐵)) ∈ V
22		foeq1 6111	. . . . . 6 ⊢ (𝑔 = (𝑓 ∩ (𝐴 × 𝐵)) → (𝑔:𝐴–onto→𝐵 ↔ (𝑓 ∩ (𝐴 × 𝐵)):𝐴–onto→𝐵))
23	21, 22	spcev 3300	. . . . 5 ⊢ ((𝑓 ∩ (𝐴 × 𝐵)):𝐴–onto→𝐵 → ∃𝑔 𝑔:𝐴–onto→𝐵)
24	19, 23	syl 17	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥) → ∃𝑔 𝑔:𝐴–onto→𝐵)
25	24	exlimiv 1858	. . 3 ⊢ (∃𝑓(∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥) → ∃𝑔 𝑔:𝐴–onto→𝐵)
26		foeq1 6111	. . . 4 ⊢ (𝑔 = 𝑓 → (𝑔:𝐴–onto→𝐵 ↔ 𝑓:𝐴–onto→𝐵))
27	26	cbvexv 2275	. . 3 ⊢ (∃𝑔 𝑔:𝐴–onto→𝐵 ↔ ∃𝑓 𝑓:𝐴–onto→𝐵)
28	25, 27	sylib 208	. 2 ⊢ (∃𝑓(∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥) → ∃𝑓 𝑓:𝐴–onto→𝐵)
29	6, 28	impbii 199	1 ⊢ (∃𝑓 𝑓:𝐴–onto→𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  ∃!wreu 2914   ∩ cin 3573   ⊆ wss 3574   class class class wbr 4653   × cxp 5112  ran crn 5115  ⟶wf 5884  –onto→wfo 5886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896

This theorem is referenced by: (None)