Theorem funsssuppss 7839

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 6343	. . . . . . . . . 10 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐺 → Fun 𝐹))
2	1	impcom 411	. . . . . . . . 9 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → Fun 𝐹)
3	2	funfnd 6355	. . . . . . . 8 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → 𝐹 Fn dom 𝐹)
4		funfn 6354	. . . . . . . . . 10 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺)
5	4	biimpi 219	. . . . . . . . 9 ⊢ (Fun 𝐺 → 𝐺 Fn dom 𝐺)
6	5	adantr 484	. . . . . . . 8 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → 𝐺 Fn dom 𝐺)
7	3, 6	jca 515	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → (𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺))
8	7	3adant3 1129	. . . . . 6 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺))
9	8	adantr 484	. . . . 5 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → (𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺))
10		dmss 5735	. . . . . . . 8 ⊢ (𝐹 ⊆ 𝐺 → dom 𝐹 ⊆ dom 𝐺)
11	10	3ad2ant2 1131	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → dom 𝐹 ⊆ dom 𝐺)
12	11	adantr 484	. . . . . 6 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → dom 𝐹 ⊆ dom 𝐺)
13		dmexg 7594	. . . . . . . 8 ⊢ (𝐺 ∈ 𝑉 → dom 𝐺 ∈ V)
14	13	3ad2ant3 1132	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → dom 𝐺 ∈ V)
15	14	adantr 484	. . . . . 6 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → dom 𝐺 ∈ V)
16		simpr 488	. . . . . 6 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
17	12, 15, 16	3jca 1125	. . . . 5 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → (dom 𝐹 ⊆ dom 𝐺 ∧ dom 𝐺 ∈ V ∧ 𝑍 ∈ V))
18	9, 17	jca 515	. . . 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 6666	. . . . . . . . 9 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝑥 ∈ dom 𝐹) → (𝐺‘𝑥) = (𝐹‘𝑥))
20	19	3expa 1115	. . . . . . . 8 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ dom 𝐹) → (𝐺‘𝑥) = (𝐹‘𝑥))
21		eqeq1 2802	. . . . . . . . 9 ⊢ ((𝐺‘𝑥) = (𝐹‘𝑥) → ((𝐺‘𝑥) = 𝑍 ↔ (𝐹‘𝑥) = 𝑍))
22	21	biimpd 232	. . . . . . . 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 3149	. . . . . 6 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → ∀𝑥 ∈ dom 𝐹((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
25	24	3adant3 1129	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → ∀𝑥 ∈ dom 𝐹((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
26	25	adantr 484	. . . 4 ⊢ (((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) ∧ 𝑍 ∈ V) → ∀𝑥 ∈ dom 𝐹((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍))
27		suppfnss 7838	. . . 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 417	. 2 ⊢ (𝑍 ∈ V → ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
30		ssid 3937	. . . 4 ⊢ ∅ ⊆ ∅
31		simpr 488	. . . . . 6 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
32		supp0prc 7816	. . . . . 6 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
33	31, 32	nsyl5 162	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
34		simpr 488	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
35		supp0prc 7816	. . . . . 6 ⊢ (¬ (𝐺 ∈ V ∧ 𝑍 ∈ V) → (𝐺 supp 𝑍) = ∅)
36	34, 35	nsyl5 162	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐺 supp 𝑍) = ∅)
37	33, 36	sseq12d 3948	. . . 4 ⊢ (¬ 𝑍 ∈ V → ((𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍) ↔ ∅ ⊆ ∅))
38	30, 37	mpbiri 261	. . 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 185	1 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  dom cdm 5519  Fun wfun 6318   Fn wfn 6319  ‘cfv 6324  (class class class)co 7135   supp csupp 7813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-supp 7814

This theorem is referenced by:  tdeglem4  24661