Theorem fiss 7140

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 𝑟 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
2		sspwb 4301	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
3		ssrin 3429	. . . . 5 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝒫 𝐴 ∩ Fin) ⊆ (𝒫 𝐵 ∩ Fin))
4	2, 3	sylbi 121	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝒫 𝐴 ∩ Fin) ⊆ (𝒫 𝐵 ∩ Fin))
5		ssrexv 3289	. . . 4 ⊢ ((𝒫 𝐴 ∩ Fin) ⊆ (𝒫 𝐵 ∩ Fin) → (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑟 = ∩ 𝑥 → ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑟 = ∩ 𝑥))
6	1, 4, 5	3syl 17	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑟 = ∩ 𝑥 → ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑟 = ∩ 𝑥))
7		vex 2802	. . . 4 ⊢ 𝑟 ∈ V
8		simpl 109	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ 𝑉)
9	8, 1	ssexd 4223	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ V)
10		elfi 7134	. . . 4 ⊢ ((𝑟 ∈ V ∧ 𝐴 ∈ V) → (𝑟 ∈ (fi‘𝐴) ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑟 = ∩ 𝑥))
11	7, 9, 10	sylancr 414	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝑟 ∈ (fi‘𝐴) ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑟 = ∩ 𝑥))
12		elfi 7134	. . . . 5 ⊢ ((𝑟 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝑟 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑟 = ∩ 𝑥))
13	7, 12	mpan 424	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑟 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑟 = ∩ 𝑥))
14	13	adantr 276	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝑟 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑟 = ∩ 𝑥))
15	6, 11, 14	3imtr4d 203	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝑟 ∈ (fi‘𝐴) → 𝑟 ∈ (fi‘𝐵)))
16	15	ssrdv 3230	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∃wrex 2509  Vcvv 2799   ∩ cin 3196   ⊆ wss 3197  𝒫 cpw 3649  ∩ cint 3922  ‘cfv 5317  Fincfn 6885  ficfi 7131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-er 6678  df-en 6886  df-fin 6888  df-fi 7132

This theorem is referenced by: (None)