Theorem fiss 9339

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 3942	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝑦 → 𝐴 ⊆ 𝑦))
2	1	adantl 481	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝐵 ⊆ 𝑦 → 𝐴 ⊆ 𝑦))
3	2	anim1d 612	. . . 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 4019	. . 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 4926	. . 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 5270	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
8	7	ancoms 458	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ V)
9		dffi2 9338	. . 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 9338	. . 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 3991	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903  ∩ cint 4904  ‘cfv 6500  ficfi 9325

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 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7819  df-1o 8407  df-2o 8408  df-en 8896  df-fin 8899  df-fi 9326

This theorem is referenced by:  fipwuni  9341  elfiun  9345  tgfiss  22947  ordtbas  23148  leordtval2  23168  lecldbas  23175  2ndcsb  23405  ptbasfi  23537  fclscmpi  23985  prdsxmslem2  24485