Theorem funsssuppss 8177

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 6566	. . . . . . . . . 10 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐺 → Fun 𝐹))
2	1	impcom 406	. . . . . . . . 9 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → Fun 𝐹)
3	2	funfnd 6578	. . . . . . . 8 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → 𝐹 Fn dom 𝐹)
4		funfn 6577	. . . . . . . . . 10 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺)
5	4	biimpi 215	. . . . . . . . 9 ⊢ (Fun 𝐺 → 𝐺 Fn dom 𝐺)
6	5	adantr 479	. . . . . . . 8 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → 𝐺 Fn dom 𝐺)
7	3, 6	jca 510	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → (𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺))
8	7	3adant3 1130	. . . . . 6 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺))
9	8	adantr 479	. . . . 5 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → (𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺))
10		dmss 5901	. . . . . . . 8 ⊢ (𝐹 ⊆ 𝐺 → dom 𝐹 ⊆ dom 𝐺)
11	10	3ad2ant2 1132	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → dom 𝐹 ⊆ dom 𝐺)
12	11	adantr 479	. . . . . 6 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → dom 𝐹 ⊆ dom 𝐺)
13		dmexg 7896	. . . . . . . 8 ⊢ (𝐺 ∈ 𝑉 → dom 𝐺 ∈ V)
14	13	3ad2ant3 1133	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → dom 𝐺 ∈ V)
15	14	adantr 479	. . . . . 6 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → dom 𝐺 ∈ V)
16		simpr 483	. . . . . 6 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
17	12, 15, 16	3jca 1126	. . . . 5 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → (dom 𝐹 ⊆ dom 𝐺 ∧ dom 𝐺 ∈ V ∧ 𝑍 ∈ V))
18	9, 17	jca 510	. . . 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)))
19		funssfv 6911	. . . . . . . . 9 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝑥 ∈ dom 𝐹) → (𝐺‘𝑥) = (𝐹‘𝑥))
20	19	3expa 1116	. . . . . . . 8 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ dom 𝐹) → (𝐺‘𝑥) = (𝐹‘𝑥))
21		eqeq1 2734	. . . . . . . . 9 ⊢ ((𝐺‘𝑥) = (𝐹‘𝑥) → ((𝐺‘𝑥) = 𝑍 ↔ (𝐹‘𝑥) = 𝑍))
22	21	biimpd 228	. . . . . . . 8 ⊢ ((𝐺‘𝑥) = (𝐹‘𝑥) → ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
23	20, 22	syl 17	. . . . . . 7 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ dom 𝐹) → ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
24	23	ralrimiva 3144	. . . . . 6 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → ∀𝑥 ∈ dom 𝐹((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
25	24	3adant3 1130	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → ∀𝑥 ∈ dom 𝐹((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
26	25	adantr 479	. . . 4 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → ∀𝑥 ∈ dom 𝐹((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
27		suppfnss 8176	. . . 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)))
28	18, 26, 27	sylc 65	. . 3 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
29	28	expcom 412	. 2 ⊢ (𝑍 ∈ V → ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
30		ssid 4003	. . . 4 ⊢ ∅ ⊆ ∅
31		simpr 483	. . . . . 6 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
32		supp0prc 8151	. . . . . 6 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
33	31, 32	nsyl5 159	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
34		simpr 483	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
35		supp0prc 8151	. . . . . 6 ⊢ (¬ (𝐺 ∈ V ∧ 𝑍 ∈ V) → (𝐺 supp 𝑍) = ∅)
36	34, 35	nsyl5 159	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐺 supp 𝑍) = ∅)
37	33, 36	sseq12d 4014	. . . 4 ⊢ (¬ 𝑍 ∈ V → ((𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍) ↔ ∅ ⊆ ∅))
38	30, 37	mpbiri 257	. . 3 ⊢ (¬ 𝑍 ∈ V → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
39	38	a1d 25	. 2 ⊢ (¬ 𝑍 ∈ V → ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
40	29, 39	pm2.61i 182	1 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  ∀wral 3059  Vcvv 3472   ⊆ wss 3947  ∅c0 4321  dom cdm 5675  Fun wfun 6536   Fn wfn 6537  ‘cfv 6542  (class class class)co 7411   supp csupp 8148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-supp 8149

This theorem is referenced by:  tdeglem4  25812  tdeglem4OLD  25813  fsuppss  41371