Theorem extmptsuppeq 8213

Description: The support of an extended function is the same as the original. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 30-Jun-2019.)

Hypotheses

Ref	Expression
extmptsuppeq.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
extmptsuppeq.a	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
extmptsuppeq.z	⊢ ((𝜑 ∧ 𝑛 ∈ (𝐵 ∖ 𝐴)) → 𝑋 = 𝑍)

Assertion

Ref	Expression
extmptsuppeq	⊢ (𝜑 → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍))

Distinct variable groups:   𝐴,𝑛   𝐵,𝑛   𝑛,𝑍   𝜑,𝑛

Allowed substitution hints:   𝑊(𝑛)   𝑋(𝑛)

Proof of Theorem extmptsuppeq

Step	Hyp	Ref	Expression
1		extmptsuppeq.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2	1	adantl 481	. . . . . . . 8 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝐴 ⊆ 𝐵)
3	2	sseld 3982	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ 𝜑) → (𝑛 ∈ 𝐴 → 𝑛 ∈ 𝐵))
4	3	anim1d 611	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐴 ∧ 𝑋 ∈ (V ∖ {𝑍})) → (𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍}))))
5		eldif 3961	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝐵 ∖ 𝐴) ↔ (𝑛 ∈ 𝐵 ∧ ¬ 𝑛 ∈ 𝐴))
6		extmptsuppeq.z	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐵 ∖ 𝐴)) → 𝑋 = 𝑍)
7	6	adantll 714	. . . . . . . . . . . . 13 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛 ∈ (𝐵 ∖ 𝐴)) → 𝑋 = 𝑍)
8	5, 7	sylan2br 595	. . . . . . . . . . . 12 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛 ∈ 𝐵 ∧ ¬ 𝑛 ∈ 𝐴)) → 𝑋 = 𝑍)
9	8	expr 456	. . . . . . . . . . 11 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛 ∈ 𝐵) → (¬ 𝑛 ∈ 𝐴 → 𝑋 = 𝑍))
10		elsn2g 4664	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ V → (𝑋 ∈ {𝑍} ↔ 𝑋 = 𝑍))
11		elndif 4133	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ {𝑍} → ¬ 𝑋 ∈ (V ∖ {𝑍}))
12	10, 11	biimtrrdi 254	. . . . . . . . . . . 12 ⊢ (𝑍 ∈ V → (𝑋 = 𝑍 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
13	12	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛 ∈ 𝐵) → (𝑋 = 𝑍 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
14	9, 13	syld 47	. . . . . . . . . 10 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛 ∈ 𝐵) → (¬ 𝑛 ∈ 𝐴 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
15	14	con4d 115	. . . . . . . . 9 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛 ∈ 𝐵) → (𝑋 ∈ (V ∖ {𝑍}) → 𝑛 ∈ 𝐴))
16	15	impr 454	. . . . . . . 8 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍}))) → 𝑛 ∈ 𝐴)
17		simprr 773	. . . . . . . 8 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍}))) → 𝑋 ∈ (V ∖ {𝑍}))
18	16, 17	jca 511	. . . . . . 7 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍}))) → (𝑛 ∈ 𝐴 ∧ 𝑋 ∈ (V ∖ {𝑍})))
19	18	ex 412	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍})) → (𝑛 ∈ 𝐴 ∧ 𝑋 ∈ (V ∖ {𝑍}))))
20	4, 19	impbid 212	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐴 ∧ 𝑋 ∈ (V ∖ {𝑍})) ↔ (𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍}))))
21	20	rabbidva2 3438	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → {𝑛 ∈ 𝐴 ∣ 𝑋 ∈ (V ∖ {𝑍})} = {𝑛 ∈ 𝐵 ∣ 𝑋 ∈ (V ∖ {𝑍})})
22		eqid 2737	. . . . 5 ⊢ (𝑛 ∈ 𝐴 ↦ 𝑋) = (𝑛 ∈ 𝐴 ↦ 𝑋)
23		extmptsuppeq.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
24	23, 1	ssexd 5324	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
25	24	adantl 481	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝐴 ∈ V)
26		simpl 482	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
27	22, 25, 26	mptsuppdifd 8211	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = {𝑛 ∈ 𝐴 ∣ 𝑋 ∈ (V ∖ {𝑍})})
28		eqid 2737	. . . . 5 ⊢ (𝑛 ∈ 𝐵 ↦ 𝑋) = (𝑛 ∈ 𝐵 ↦ 𝑋)
29	23	adantl 481	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝐵 ∈ 𝑊)
30	28, 29, 26	mptsuppdifd 8211	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍) = {𝑛 ∈ 𝐵 ∣ 𝑋 ∈ (V ∖ {𝑍})})
31	21, 27, 30	3eqtr4d 2787	. . 3 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍))
32	31	ex 412	. 2 ⊢ (𝑍 ∈ V → (𝜑 → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍)))
33		simpr 484	. . . . 5 ⊢ (((𝑛 ∈ 𝐴 ↦ 𝑋) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
34		supp0prc 8188	. . . . 5 ⊢ (¬ ((𝑛 ∈ 𝐴 ↦ 𝑋) ∈ V ∧ 𝑍 ∈ V) → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ∅)
35	33, 34	nsyl5 159	. . . 4 ⊢ (¬ 𝑍 ∈ V → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ∅)
36		simpr 484	. . . . 5 ⊢ (((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
37		supp0prc 8188	. . . . 5 ⊢ (¬ ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V ∧ 𝑍 ∈ V) → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍) = ∅)
38	36, 37	nsyl5 159	. . . 4 ⊢ (¬ 𝑍 ∈ V → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍) = ∅)
39	35, 38	eqtr4d 2780	. . 3 ⊢ (¬ 𝑍 ∈ V → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍))
40	39	a1d 25	. 2 ⊢ (¬ 𝑍 ∈ V → (𝜑 → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍)))
41	32, 40	pm2.61i 182	1 ⊢ (𝜑 → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {crab 3436  Vcvv 3480   ∖ cdif 3948   ⊆ wss 3951  ∅c0 4333  {csn 4626   ↦ cmpt 5225  (class class class)co 7431   supp csupp 8185

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186

This theorem is referenced by:  cantnfrescl  9716  cantnfres  9717