Theorem exfo 7090

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 7088	. . . 4 ⊢ (𝑓:𝐴–onto→𝐵 ↔ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥))
2		dff4 7086	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 ↔ (𝑓 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦))
3	2	simprbi 502	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦)
4	3	anim1i 626	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥) → (∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥))
5	1, 4	sylbi 220	. . 3 ⊢ (𝑓:𝐴–onto→𝐵 → (∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥))
6	5	eximi 1858	. 2 ⊢ (∃𝑓 𝑓:𝐴–onto→𝐵 → ∃𝑓(∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥))
7		brinxp 5731	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑓𝑦 ↔ 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
8	7	reubidva 3384	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ↔ ∃!𝑦 ∈ 𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
9	8	biimpd 232	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 → ∃!𝑦 ∈ 𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
10	9	ralimia 3099	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 → ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦)
11		inss2 4192	. . . . . . . . 9 ⊢ (𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
12	10, 11	jctil 528	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 → ((𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
13		dff4 7086	. . . . . . . 8 ⊢ ((𝑓 ∩ (𝐴 × 𝐵)):𝐴⟶𝐵 ↔ ((𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
14	12, 13	sylibr 237	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 → (𝑓 ∩ (𝐴 × 𝐵)):𝐴⟶𝐵)
15		rninxp 6169	. . . . . . . 8 ⊢ (ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥)
16	15	biimpri 231	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥 → ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵)
17	14, 16	anim12i 624	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥) → ((𝑓 ∩ (𝐴 × 𝐵)):𝐴⟶𝐵 ∧ ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵))
18		dffo2 6786	. . . . . 6 ⊢ ((𝑓 ∩ (𝐴 × 𝐵)):𝐴–onto→𝐵 ↔ ((𝑓 ∩ (𝐴 × 𝐵)):𝐴⟶𝐵 ∧ ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵))
19	17, 18	sylibr 237	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥) → (𝑓 ∩ (𝐴 × 𝐵)):𝐴–onto→𝐵)
20		vex 3461	. . . . . . 7 ⊢ 𝑓 ∈ V
21	20	inex1 5278	. . . . . 6 ⊢ (𝑓 ∩ (𝐴 × 𝐵)) ∈ V
22		foeq1 6778	. . . . . 6 ⊢ (𝑔 = (𝑓 ∩ (𝐴 × 𝐵)) → (𝑔:𝐴–onto→𝐵 ↔ (𝑓 ∩ (𝐴 × 𝐵)):𝐴–onto→𝐵))
23	21, 22	spcev 3568	. . . . 5 ⊢ ((𝑓 ∩ (𝐴 × 𝐵)):𝐴–onto→𝐵 → ∃𝑔 𝑔:𝐴–onto→𝐵)
24	19, 23	syl 18	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥) → ∃𝑔 𝑔:𝐴–onto→𝐵)
25	24	exlimiv 1953	. . 3 ⊢ (∃𝑓(∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥) → ∃𝑔 𝑔:𝐴–onto→𝐵)
26		foeq1 6778	. . . 4 ⊢ (𝑔 = 𝑓 → (𝑔:𝐴–onto→𝐵 ↔ 𝑓:𝐴–onto→𝐵))
27	26	cbvexvw 2060	. . 3 ⊢ (∃𝑔 𝑔:𝐴–onto→𝐵 ↔ ∃𝑓 𝑓:𝐴–onto→𝐵)
28	25, 27	sylib 221	. 2 ⊢ (∃𝑓(∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥) → ∃𝑓 𝑓:𝐴–onto→𝐵)
29	6, 28	impbii 212	1 ⊢ (∃𝑓 𝑓:𝐴–onto→𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  ∃!wreu 3368   ∩ cin 3906   ⊆ wss 3907   class class class wbr 5105   × cxp 5650  ran crn 5653  ⟶wf 6521  –onto→wfo 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fo 6531  df-fv 6533

This theorem is referenced by: (None)