Theorem funsssuppss 8188

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 6552	. . . . . . . . . 10 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐺 → Fun 𝐹))
2	1	impcom 413	. . . . . . . . 9 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → Fun 𝐹)
3	2	funfnd 6564	. . . . . . . 8 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → 𝐹 Fn dom 𝐹)
4		funfn 6563	. . . . . . . . 9 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺)
5	4	birani 509	. . . . . . . 8 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → 𝐺 Fn dom 𝐺)
6	3, 5	jca 521	. . . . . . 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 486	. . . . 5 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → (𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺))
9		dmss 5886	. . . . . . . 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 486	. . . . . 6 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → dom 𝐹 ⊆ dom 𝐺)
12		dmexg 7898	. . . . . . . 8 ⊢ (𝐺 ∈ 𝑉 → dom 𝐺 ∈ V)
13	12	3ad2ant3 1153	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → dom 𝐺 ∈ V)
14	13	adantr 486	. . . . . 6 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → dom 𝐺 ∈ V)
15		simpr 490	. . . . . 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 521	. . . 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 6899	. . . . . . . . 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 2764	. . . . . . . . 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 3154	. . . . . 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 486	. . . 4 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → ∀𝑥 ∈ dom 𝐹((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
26		suppfnss 8187	. . . 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 419	. 2 ⊢ (𝑍 ∈ V → ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
29		ssid 3953	. . . 4 ⊢ ∅ ⊆ ∅
30		simpr 490	. . . . . 6 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
31		supp0prc 8161	. . . . . 6 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
32	30, 31	nsyl5 160	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
33		simpr 490	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
34		supp0prc 8161	. . . . . 6 ⊢ (¬ (𝐺 ∈ V ∧ 𝑍 ∈ V) → (𝐺 supp 𝑍) = ∅)
35	33, 34	nsyl5 160	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐺 supp 𝑍) = ∅)
36	32, 35	sseq12d 3964	. . . 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ⊆ wss 3899  ∅c0 4279  dom cdm 5655  Fun wfun 6527   Fn wfn 6528  ‘cfv 6533  (class class class)co 7413   supp csupp 8158

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-supp 8159

This theorem is used by:  fsuppss  9353  tdeglem4  26285