Theorem extmptsuppeq 8128

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 3936	. . . . . . 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 3915	. . . . . . . . . . . . 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 4618	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ V → (𝑋 ∈ {𝑍} ↔ 𝑋 = 𝑍))
11		elndif 4086	. . . . . . . . . . . . 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 772	. . . . . . . 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 3398	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → {𝑛 ∈ 𝐴 ∣ 𝑋 ∈ (V ∖ {𝑍})} = {𝑛 ∈ 𝐵 ∣ 𝑋 ∈ (V ∖ {𝑍})})
22		eqid 2729	. . . . 5 ⊢ (𝑛 ∈ 𝐴 ↦ 𝑋) = (𝑛 ∈ 𝐴 ↦ 𝑋)
23		extmptsuppeq.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
24	23, 1	ssexd 5266	. . . . . 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 8126	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = {𝑛 ∈ 𝐴 ∣ 𝑋 ∈ (V ∖ {𝑍})})
28		eqid 2729	. . . . 5 ⊢ (𝑛 ∈ 𝐵 ↦ 𝑋) = (𝑛 ∈ 𝐵 ↦ 𝑋)
29	23	adantl 481	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝐵 ∈ 𝑊)
30	28, 29, 26	mptsuppdifd 8126	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍) = {𝑛 ∈ 𝐵 ∣ 𝑋 ∈ (V ∖ {𝑍})})
31	21, 27, 30	3eqtr4d 2774	. . 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 8103	. . . . 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 8103	. . . . 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 2767	. . 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 2109  {crab 3396  Vcvv 3438   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286  {csn 4579   ↦ cmpt 5176  (class class class)co 7353   supp csupp 8100

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-supp 8101

This theorem is referenced by:  cantnfrescl  9591  cantnfres  9592