Theorem extmptsuppeq 8236

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 487	. . . . . . . 8 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝐴 ⊆ 𝐵)
3	2	sseld 3912	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ 𝜑) → (𝑛 ∈ 𝐴 → 𝑛 ∈ 𝐵))
4	3	anim1d 623	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐴 ∧ 𝑋 ∈ (V ∖ {𝑍})) → (𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍}))))
5		eldif 4052	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝐵 ∖ 𝐴) ↔ (𝑛 ∈ 𝐵 ∧ ¬ 𝑛 ∈ 𝐴))
6		extmptsuppeq.z	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐵 ∖ 𝐴)) → 𝑋 = 𝑍)
7	6	adantll 727	. . . . . . . . . . . . 13 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛 ∈ (𝐵 ∖ 𝐴)) → 𝑋 = 𝑍)
8	5, 7	sylan2br 607	. . . . . . . . . . . 12 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛 ∈ 𝐵 ∧ ¬ 𝑛 ∈ 𝐴)) → 𝑋 = 𝑍)
9	8	expr 462	. . . . . . . . . . 11 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛 ∈ 𝐵) → (¬ 𝑛 ∈ 𝐴 → 𝑋 = 𝑍))
10		elsn2g 4625	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ V → (𝑋 ∈ {𝑍} ↔ 𝑋 = 𝑍))
11		elndif 4072	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ {𝑍} → ¬ 𝑋 ∈ (V ∖ {𝑍}))
12	10, 11	biimtrrdi 257	. . . . . . . . . . . 12 ⊢ (𝑍 ∈ V → (𝑋 = 𝑍 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
13	12	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛 ∈ 𝐵) → (𝑋 = 𝑍 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
14	9, 13	syld 48	. . . . . . . . . 10 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛 ∈ 𝐵) → (¬ 𝑛 ∈ 𝐴 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
15	14	con4d 116	. . . . . . . . 9 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛 ∈ 𝐵) → (𝑋 ∈ (V ∖ {𝑍}) → 𝑛 ∈ 𝐴))
16	15	impr 460	. . . . . . . 8 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍}))) → 𝑛 ∈ 𝐴)
17		simprr 785	. . . . . . . 8 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍}))) → 𝑋 ∈ (V ∖ {𝑍}))
18	16, 17	jca 521	. . . . . . 7 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍}))) → (𝑛 ∈ 𝐴 ∧ 𝑋 ∈ (V ∖ {𝑍})))
19	18	ex 418	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍})) → (𝑛 ∈ 𝐴 ∧ 𝑋 ∈ (V ∖ {𝑍}))))
20	4, 19	impbid 215	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐴 ∧ 𝑋 ∈ (V ∖ {𝑍})) ↔ (𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍}))))
21	20	rabbidva2 3418	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → {𝑛 ∈ 𝐴 ∣ 𝑋 ∈ (V ∖ {𝑍})} = {𝑛 ∈ 𝐵 ∣ 𝑋 ∈ (V ∖ {𝑍})})
22		eqid 2763	. . . . 5 ⊢ (𝑛 ∈ 𝐴 ↦ 𝑋) = (𝑛 ∈ 𝐴 ↦ 𝑋)
23		extmptsuppeq.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
24	23, 1	ssexd 5291	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
25	24	adantl 487	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝐴 ∈ V)
26		simpl 488	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
27	22, 25, 26	mptsuppdifd 8234	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = {𝑛 ∈ 𝐴 ∣ 𝑋 ∈ (V ∖ {𝑍})})
28		eqid 2763	. . . . 5 ⊢ (𝑛 ∈ 𝐵 ↦ 𝑋) = (𝑛 ∈ 𝐵 ↦ 𝑋)
29	23	adantl 487	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝐵 ∈ 𝑊)
30	28, 29, 26	mptsuppdifd 8234	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍) = {𝑛 ∈ 𝐵 ∣ 𝑋 ∈ (V ∖ {𝑍})})
31	21, 27, 30	3eqtr4d 2808	. . 3 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍))
32	31	ex 418	. 2 ⊢ (𝑍 ∈ V → (𝜑 → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍)))
33		simpr 490	. . . . 5 ⊢ (((𝑛 ∈ 𝐴 ↦ 𝑋) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
34		supp0prc 8210	. . . . 5 ⊢ (¬ ((𝑛 ∈ 𝐴 ↦ 𝑋) ∈ V ∧ 𝑍 ∈ V) → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ∅)
35	33, 34	nsyl5 160	. . . 4 ⊢ (¬ 𝑍 ∈ V → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ∅)
36		simpr 490	. . . . 5 ⊢ (((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
37		supp0prc 8210	. . . . 5 ⊢ (¬ ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V ∧ 𝑍 ∈ V) → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍) = ∅)
38	36, 37	nsyl5 160	. . . 4 ⊢ (¬ 𝑍 ∈ V → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍) = ∅)
39	35, 38	eqtr4d 2801	. . 3 ⊢ (¬ 𝑍 ∈ V → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍))
40	39	a1d 26	. 2 ⊢ (¬ 𝑍 ∈ V → (𝜑 → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍)))
41	32, 40	pm2.61i 184	1 ⊢ (𝜑 → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048  ∅c0 4281  {csn 4588   ↦ cmpt 5190  (class class class)co 7446   supp csupp 8207

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-supp 8208

This theorem is used by:  cantnfrescl  9707  cantnfres  9708