Theorem fiss 8737

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 3898	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝑦 → 𝐴 ⊆ 𝑦))
2	1	adantl 482	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝐵 ⊆ 𝑦 → 𝐴 ⊆ 𝑦))
3	2	anim1d 610	. . . 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 3967	. . 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 4805	. . 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 5121	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
8	7	ancoms 459	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ V)
9		dffi2 8736	. . 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 8736	. . 3 ⊢ (𝐵 ∈ 𝑉 → (fi‘𝐵) = ∩ {𝑦 ∣ (𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)})
12	11	adantr 481	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐵) = ∩ {𝑦 ∣ (𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)})
13	6, 10, 12	3sstr4d 3937	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2080  {cab 2774  ∀wral 3104  Vcvv 3436   ∩ cin 3860   ⊆ wss 3861  ∩ cint 4784  ‘cfv 6228  ficfi 8723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-reu 3111  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-int 4785  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-pred 6026  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-ov 7022  df-oprab 7023  df-mpo 7024  df-om 7440  df-wrecs 7801  df-recs 7863  df-rdg 7901  df-1o 7956  df-oadd 7960  df-er 8142  df-en 8361  df-fin 8364  df-fi 8724

This theorem is referenced by:  fipwuni  8739  elfiun  8743  tgfiss  21283  ordtbas  21484  leordtval2  21504  lecldbas  21511  2ndcsb  21741  ptbasfi  21873  fclscmpi  22321  prdsxmslem2  22822