Theorem funsssuppss 8187

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 variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funss 6557	. . . . . . . . . 10 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐺 → Fun 𝐹))
2	1	impcom 412	. . . . . . . . 9 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → Fun 𝐹)
3	2	funfnd 6569	. . . . . . . 8 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → 𝐹 Fn dom 𝐹)
4		funfn 6568	. . . . . . . . 9 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺)
5	4	birani 508	. . . . . . . 8 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → 𝐺 Fn dom 𝐺)
6	3, 5	jca 520	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → (𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺))
7	6	3adant3 1150	. . . . . 6 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺))
8	7	adantr 485	. . . . 5 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → (𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺))
9		dmss 5894	. . . . . . . 8 ⊢ (𝐹 ⊆ 𝐺 → dom 𝐹 ⊆ dom 𝐺)
10	9	3ad2ant2 1152	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → dom 𝐹 ⊆ dom 𝐺)
11	10	adantr 485	. . . . . 6 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → dom 𝐹 ⊆ dom 𝐺)
12		dmexg 7899	. . . . . . . 8 ⊢ (𝐺 ∈ 𝑉 → dom 𝐺 ∈ V)
13	12	3ad2ant3 1153	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → dom 𝐺 ∈ V)
14	13	adantr 485	. . . . . 6 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → dom 𝐺 ∈ V)
15		simpr 489	. . . . . 6 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
16	11, 14, 15	3jca 1146	. . . . 5 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → (dom 𝐹 ⊆ dom 𝐺 ∧ dom 𝐺 ∈ V ∧ 𝑍 ∈ V))
17	8, 16	jca 520	. . . 4 ⊢ (((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)))
18		funssfv 6904	. . . . . . . . 9 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝑥 ∈ dom 𝐹) → (𝐺‘𝑥) = (𝐹‘𝑥))
19	18	3expa 1136	. . . . . . . 8 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ dom 𝐹) → (𝐺‘𝑥) = (𝐹‘𝑥))
20		eqeq1 2767	. . . . . . . . 9 ⊢ ((𝐺‘𝑥) = (𝐹‘𝑥) → ((𝐺‘𝑥) = 𝑍 ↔ (𝐹‘𝑥) = 𝑍))
21	20	biimpd 232	. . . . . . . 8 ⊢ ((𝐺‘𝑥) = (𝐹‘𝑥) → ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
22	19, 21	syl 18	. . . . . . 7 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ dom 𝐹) → ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
23	22	ralrimiva 3157	. . . . . 6 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → ∀𝑥 ∈ dom 𝐹((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
24	23	3adant3 1150	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → ∀𝑥 ∈ dom 𝐹((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
25	24	adantr 485	. . . 4 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → ∀𝑥 ∈ dom 𝐹((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
26		suppfnss 8186	. . . 4 ⊢ (((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)))
27	17, 25, 26	sylc 66	. . 3 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
28	27	expcom 418	. 2 ⊢ (𝑍 ∈ V → ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
29		ssid 3960	. . . 4 ⊢ ∅ ⊆ ∅
30		simpr 489	. . . . . 6 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
31		supp0prc 8160	. . . . . 6 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
32	30, 31	nsyl5 160	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
33		simpr 489	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
34		supp0prc 8160	. . . . . 6 ⊢ (¬ (𝐺 ∈ V ∧ 𝑍 ∈ V) → (𝐺 supp 𝑍) = ∅)
35	33, 34	nsyl5 160	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐺 supp 𝑍) = ∅)
36	32, 35	sseq12d 3971	. . . 4 ⊢ (¬ 𝑍 ∈ V → ((𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍) ↔ ∅ ⊆ ∅))
37	29, 36	mpbiri 261	. . 3 ⊢ (¬ 𝑍 ∈ V → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
38	37	a1d 26	. 2 ⊢ (¬ 𝑍 ∈ V → ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
39	28, 38	pm2.61i 184	1 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ⊆ wss 3906  ∅c0 4287  dom cdm 5663  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538  (class class class)co 7412   supp csupp 8157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-supp 8158

This theorem is referenced by:  fsuppss  9344  tdeglem4  26198