Theorem extmptsuppeq 8135

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 482	. . . . . . . 8 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝐴 ⊆ 𝐵)
3	2	sseld 3921	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ 𝜑) → (𝑛 ∈ 𝐴 → 𝑛 ∈ 𝐵))
4	3	anim1d 617	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐴 ∧ 𝑋 ∈ (V ∖ {𝑍})) → (𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍}))))
5		eldif 3900	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝐵 ∖ 𝐴) ↔ (𝑛 ∈ 𝐵 ∧ ¬ 𝑛 ∈ 𝐴))
6		extmptsuppeq.z	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐵 ∖ 𝐴)) → 𝑋 = 𝑍)
7	6	adantll 720	. . . . . . . . . . . . 13 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛 ∈ (𝐵 ∖ 𝐴)) → 𝑋 = 𝑍)
8	5, 7	sylan2br 601	. . . . . . . . . . . 12 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛 ∈ 𝐵 ∧ ¬ 𝑛 ∈ 𝐴)) → 𝑋 = 𝑍)
9	8	expr 457	. . . . . . . . . . 11 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛 ∈ 𝐵) → (¬ 𝑛 ∈ 𝐴 → 𝑋 = 𝑍))
10		elsn2g 4603	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ V → (𝑋 ∈ {𝑍} ↔ 𝑋 = 𝑍))
11		elndif 4070	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ {𝑍} → ¬ 𝑋 ∈ (V ∖ {𝑍}))
12	10, 11	biimtrrdi 255	. . . . . . . . . . . 12 ⊢ (𝑍 ∈ V → (𝑋 = 𝑍 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
13	12	ad2antrr 732	. . . . . . . . . . 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 455	. . . . . . . 8 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍}))) → 𝑛 ∈ 𝐴)
17		simprr 778	. . . . . . . 8 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍}))) → 𝑋 ∈ (V ∖ {𝑍}))
18	16, 17	jca 516	. . . . . . 7 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍}))) → (𝑛 ∈ 𝐴 ∧ 𝑋 ∈ (V ∖ {𝑍})))
19	18	ex 413	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍})) → (𝑛 ∈ 𝐴 ∧ 𝑋 ∈ (V ∖ {𝑍}))))
20	4, 19	impbid 213	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐴 ∧ 𝑋 ∈ (V ∖ {𝑍})) ↔ (𝑛 ∈ 𝐵 ∧ 𝑋 ∈ (V ∖ {𝑍}))))
21	20	rabbidva2 3394	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → {𝑛 ∈ 𝐴 ∣ 𝑋 ∈ (V ∖ {𝑍})} = {𝑛 ∈ 𝐵 ∣ 𝑋 ∈ (V ∖ {𝑍})})
22		eqid 2740	. . . . 5 ⊢ (𝑛 ∈ 𝐴 ↦ 𝑋) = (𝑛 ∈ 𝐴 ↦ 𝑋)
23		extmptsuppeq.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
24	23, 1	ssexd 5259	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
25	24	adantl 482	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝐴 ∈ V)
26		simpl 483	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
27	22, 25, 26	mptsuppdifd 8133	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = {𝑛 ∈ 𝐴 ∣ 𝑋 ∈ (V ∖ {𝑍})})
28		eqid 2740	. . . . 5 ⊢ (𝑛 ∈ 𝐵 ↦ 𝑋) = (𝑛 ∈ 𝐵 ↦ 𝑋)
29	23	adantl 482	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝐵 ∈ 𝑊)
30	28, 29, 26	mptsuppdifd 8133	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍) = {𝑛 ∈ 𝐵 ∣ 𝑋 ∈ (V ∖ {𝑍})})
31	21, 27, 30	3eqtr4d 2785	. . 3 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍))
32	31	ex 413	. 2 ⊢ (𝑍 ∈ V → (𝜑 → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍)))
33		simpr 485	. . . . 5 ⊢ (((𝑛 ∈ 𝐴 ↦ 𝑋) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
34		supp0prc 8110	. . . . 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 485	. . . . 5 ⊢ (((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
37		supp0prc 8110	. . . . 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 2778	. . 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 183	1 ⊢ (𝜑 → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3392  Vcvv 3432   ∖ cdif 3887   ⊆ wss 3890  ∅c0 4268  {csn 4562   ↦ cmpt 5160  (class class class)co 7363   supp csupp 8107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-supp 8108

This theorem is referenced by:  cantnfrescl  9595  cantnfres  9596