Theorem fiss 9319

Description: Subset relationship for function fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Assertion

Ref	Expression
fiss	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵))

Proof of Theorem fiss

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 3937	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝑦 → 𝐴 ⊆ 𝑦))
2	1	adantl 481	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝐵 ⊆ 𝑦 → 𝐴 ⊆ 𝑦))
3	2	anim1d 611	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → ((𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦) → (𝐴 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)))
4	3	ss2abdv 4014	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → {𝑦 ∣ (𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)} ⊆ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)})
5		intss 4921	. . 3 ⊢ ({𝑦 ∣ (𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)} ⊆ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)} → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)} ⊆ ∩ {𝑦 ∣ (𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)})
6	4, 5	syl 17	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)} ⊆ ∩ {𝑦 ∣ (𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)})
7		ssexg 5265	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
8	7	ancoms 458	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ V)
9		dffi2 9318	. . 3 ⊢ (𝐴 ∈ V → (fi‘𝐴) = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)})
10	8, 9	syl 17	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)})
11		dffi2 9318	. . 3 ⊢ (𝐵 ∈ 𝑉 → (fi‘𝐵) = ∩ {𝑦 ∣ (𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)})
12	11	adantr 480	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐵) = ∩ {𝑦 ∣ (𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)})
13	6, 10, 12	3sstr4d 3986	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2711  ∀wral 3048  Vcvv 3437   ∩ cin 3897   ⊆ wss 3898  ∩ cint 4899  ‘cfv 6489  ficfi 9305

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-om 7806  df-1o 8394  df-2o 8395  df-en 8880  df-fin 8883  df-fi 9306

This theorem is referenced by:  fipwuni  9321  elfiun  9325  tgfiss  22926  ordtbas  23127  leordtval2  23147  lecldbas  23154  2ndcsb  23384  ptbasfi  23516  fclscmpi  23964  prdsxmslem2  24464