Theorem fifo 9338

Description: Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.)

Hypothesis

Ref	Expression
fifo.1	⊢ 𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦)

Assertion

Ref	Expression
fifo	⊢ (𝐴 ∈ 𝑉 → 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))

Distinct variable groups:   𝑦,𝐴   𝑦,𝑉

Allowed substitution hint:   𝐹(𝑦)

Proof of Theorem fifo

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifsni 4734	. . . . . 6 ⊢ (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑦 ≠ ∅)
2		intex 5281	. . . . . 6 ⊢ (𝑦 ≠ ∅ ↔ ∩ 𝑦 ∈ V)
3	1, 2	sylib 218	. . . . 5 ⊢ (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → ∩ 𝑦 ∈ V)
4	3	rgen 3054	. . . 4 ⊢ ∀𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})∩ 𝑦 ∈ V
5		fifo.1	. . . . 5 ⊢ 𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦)
6	5	fnmpt 6632	. . . 4 ⊢ (∀𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})∩ 𝑦 ∈ V → 𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}))
7	4, 6	mp1i 13	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}))
8		dffn4 6752	. . 3 ⊢ (𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹)
9	7, 8	sylib 218	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹)
10		elfi2 9320	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (fi‘𝐴) ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = ∩ 𝑦))
11	5	elrnmpt 5907	. . . . . 6 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = ∩ 𝑦))
12	11	elv 3435	. . . . 5 ⊢ (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = ∩ 𝑦)
13	10, 12	bitr4di 289	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (fi‘𝐴) ↔ 𝑥 ∈ ran 𝐹))
14	13	eqrdv 2735	. . 3 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ran 𝐹)
15		foeq3 6744	. . 3 ⊢ ((fi‘𝐴) = ran 𝐹 → (𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹))
16	14, 15	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹))
17	9, 16	mpbird 257	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∖ cdif 3887   ∩ cin 3889  ∅c0 4274  𝒫 cpw 4542  {csn 4568  ∩ cint 4890   ↦ cmpt 5167  ran crn 5625   Fn wfn 6487  –onto→wfo 6490  ‘cfv 6492  Fincfn 8886  ficfi 9316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-fo 6498  df-fv 6500  df-fi 9317

This theorem is referenced by:  inffien  9976  fictb  10157