Theorem funsssuppss 6457

Description: The support of a function which is a subset of another function is a subset of the support of this other function. (Contributed by AV, 27-Jul-2019.)

Assertion

Ref	Expression
funsssuppss	⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))

Proof of Theorem funsssuppss

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

Step	Hyp	Ref	Expression
1		df-supp 6435	. . . . . 6 ⊢ supp = (𝑓 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑓 ∣ (𝑓 “ {𝑖}) ≠ {𝑧}})
2	1	elmpocl2 6250	. . . . 5 ⊢ (𝑥 ∈ (𝐹 supp 𝑍) → 𝑍 ∈ V)
3		funss 5370	. . . . . . . . . . . 12 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐺 → Fun 𝐹))
4	3	impcom 125	. . . . . . . . . . 11 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → Fun 𝐹)
5	4	funfnd 5382	. . . . . . . . . 10 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → 𝐹 Fn dom 𝐹)
6		funfn 5381	. . . . . . . . . . . 12 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺)
7	6	biimpi 120	. . . . . . . . . . 11 ⊢ (Fun 𝐺 → 𝐺 Fn dom 𝐺)
8	7	adantr 276	. . . . . . . . . 10 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → 𝐺 Fn dom 𝐺)
9	5, 8	jca 306	. . . . . . . . 9 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → (𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺))
10	9	3adant3 1044	. . . . . . . 8 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺))
11	10	adantr 276	. . . . . . 7 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → (𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺))
12		dmss 4954	. . . . . . . . . 10 ⊢ (𝐹 ⊆ 𝐺 → dom 𝐹 ⊆ dom 𝐺)
13	12	3ad2ant2 1046	. . . . . . . . 9 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → dom 𝐹 ⊆ dom 𝐺)
14	13	adantr 276	. . . . . . . 8 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → dom 𝐹 ⊆ dom 𝐺)
15		dmexg 5020	. . . . . . . . . 10 ⊢ (𝐺 ∈ 𝑉 → dom 𝐺 ∈ V)
16	15	3ad2ant3 1047	. . . . . . . . 9 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → dom 𝐺 ∈ V)
17	16	adantr 276	. . . . . . . 8 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → dom 𝐺 ∈ V)
18		simpr 110	. . . . . . . 8 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
19	14, 17, 18	3jca 1204	. . . . . . 7 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → (dom 𝐹 ⊆ dom 𝐺 ∧ dom 𝐺 ∈ V ∧ 𝑍 ∈ V))
20	11, 19	jca 306	. . . . . 6 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → ((𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺) ∧ (dom 𝐹 ⊆ dom 𝐺 ∧ dom 𝐺 ∈ V ∧ 𝑍 ∈ V)))
21		funssfv 5695	. . . . . . . . . . 11 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝑥 ∈ dom 𝐹) → (𝐺‘𝑥) = (𝐹‘𝑥))
22	21	3expa 1230	. . . . . . . . . 10 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ dom 𝐹) → (𝐺‘𝑥) = (𝐹‘𝑥))
23		eqeq1 2239	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑥) = (𝐹‘𝑥) → ((𝐺‘𝑥) = 𝑍 ↔ (𝐹‘𝑥) = 𝑍))
24	23	biimpd 144	. . . . . . . . . 10 ⊢ ((𝐺‘𝑥) = (𝐹‘𝑥) → ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
25	22, 24	syl 14	. . . . . . . . 9 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ dom 𝐹) → ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
26	25	ralrimiva 2615	. . . . . . . 8 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → ∀𝑥 ∈ dom 𝐹((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
27	26	3adant3 1044	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → ∀𝑥 ∈ dom 𝐹((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
28	27	adantr 276	. . . . . 6 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → ∀𝑥 ∈ dom 𝐹((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
29		suppfnss 6456	. . . . . 6 ⊢ (((𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺) ∧ (dom 𝐹 ⊆ dom 𝐺 ∧ dom 𝐺 ∈ V ∧ 𝑍 ∈ V)) → (∀𝑥 ∈ dom 𝐹((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
30	20, 28, 29	sylc 62	. . . . 5 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
31	2, 30	sylan2 286	. . . 4 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
32		simpr 110	. . . 4 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → 𝑥 ∈ (𝐹 supp 𝑍))
33	31, 32	sseldd 3238	. . 3 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → 𝑥 ∈ (𝐺 supp 𝑍))
34	33	ex 115	. 2 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝑥 ∈ (𝐹 supp 𝑍) → 𝑥 ∈ (𝐺 supp 𝑍)))
35	34	ssrdv 3243	1 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203   ≠ wne 2412  ∀wral 2520  {crab 2524  Vcvv 2812   ⊆ wss 3210  {csn 3688  dom cdm 4748   “ cima 4751  Fun wfun 5345   Fn wfn 5346  ‘cfv 5351  (class class class)co 6049   supp csupp 6434

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-supp 6435

This theorem is referenced by: (None)