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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 (𝜑𝐵𝑊)
extmptsuppeq.a (𝜑𝐴𝐵)
extmptsuppeq.z ((𝜑𝑛 ∈ (𝐵𝐴)) → 𝑋 = 𝑍)
Assertion
Ref Expression
extmptsuppeq (𝜑 → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍))
Distinct variable groups:   𝐴,𝑛   𝐵,𝑛   𝑛,𝑍   𝜑,𝑛
Allowed substitution hints:   𝑊(𝑛)   𝑋(𝑛)

Proof of Theorem extmptsuppeq
StepHypRef Expression
1 extmptsuppeq.a . . . . . . . . 9 (𝜑𝐴𝐵)
21adantl 481 . . . . . . . 8 ((𝑍 ∈ V ∧ 𝜑) → 𝐴𝐵)
32sseld 3982 . . . . . . 7 ((𝑍 ∈ V ∧ 𝜑) → (𝑛𝐴𝑛𝐵))
43anim1d 611 . . . . . 6 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋 ∈ (V ∖ {𝑍})) → (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))))
5 eldif 3961 . . . . . . . . . . . . 13 (𝑛 ∈ (𝐵𝐴) ↔ (𝑛𝐵 ∧ ¬ 𝑛𝐴))
6 extmptsuppeq.z . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝐵𝐴)) → 𝑋 = 𝑍)
76adantll 714 . . . . . . . . . . . . 13 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛 ∈ (𝐵𝐴)) → 𝑋 = 𝑍)
85, 7sylan2br 595 . . . . . . . . . . . 12 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵 ∧ ¬ 𝑛𝐴)) → 𝑋 = 𝑍)
98expr 456 . . . . . . . . . . 11 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛𝐵) → (¬ 𝑛𝐴𝑋 = 𝑍))
10 elsn2g 4664 . . . . . . . . . . . . 13 (𝑍 ∈ V → (𝑋 ∈ {𝑍} ↔ 𝑋 = 𝑍))
11 elndif 4133 . . . . . . . . . . . . 13 (𝑋 ∈ {𝑍} → ¬ 𝑋 ∈ (V ∖ {𝑍}))
1210, 11biimtrrdi 254 . . . . . . . . . . . 12 (𝑍 ∈ V → (𝑋 = 𝑍 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
1312ad2antrr 726 . . . . . . . . . . 11 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛𝐵) → (𝑋 = 𝑍 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
149, 13syld 47 . . . . . . . . . 10 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛𝐵) → (¬ 𝑛𝐴 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
1514con4d 115 . . . . . . . . 9 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛𝐵) → (𝑋 ∈ (V ∖ {𝑍}) → 𝑛𝐴))
1615impr 454 . . . . . . . 8 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))) → 𝑛𝐴)
17 simprr 773 . . . . . . . 8 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))) → 𝑋 ∈ (V ∖ {𝑍}))
1816, 17jca 511 . . . . . . 7 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))) → (𝑛𝐴𝑋 ∈ (V ∖ {𝑍})))
1918ex 412 . . . . . 6 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐵𝑋 ∈ (V ∖ {𝑍})) → (𝑛𝐴𝑋 ∈ (V ∖ {𝑍}))))
204, 19impbid 212 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋 ∈ (V ∖ {𝑍})) ↔ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))))
2120rabbidva2 3438 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → {𝑛𝐴𝑋 ∈ (V ∖ {𝑍})} = {𝑛𝐵𝑋 ∈ (V ∖ {𝑍})})
22 eqid 2737 . . . . 5 (𝑛𝐴𝑋) = (𝑛𝐴𝑋)
23 extmptsuppeq.b . . . . . . 7 (𝜑𝐵𝑊)
2423, 1ssexd 5324 . . . . . 6 (𝜑𝐴 ∈ V)
2524adantl 481 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → 𝐴 ∈ V)
26 simpl 482 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
2722, 25, 26mptsuppdifd 8211 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋) supp 𝑍) = {𝑛𝐴𝑋 ∈ (V ∖ {𝑍})})
28 eqid 2737 . . . . 5 (𝑛𝐵𝑋) = (𝑛𝐵𝑋)
2923adantl 481 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → 𝐵𝑊)
3028, 29, 26mptsuppdifd 8211 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐵𝑋) supp 𝑍) = {𝑛𝐵𝑋 ∈ (V ∖ {𝑍})})
3121, 27, 303eqtr4d 2787 . . 3 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍))
3231ex 412 . 2 (𝑍 ∈ V → (𝜑 → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍)))
33 simpr 484 . . . . 5 (((𝑛𝐴𝑋) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
34 supp0prc 8188 . . . . 5 (¬ ((𝑛𝐴𝑋) ∈ V ∧ 𝑍 ∈ V) → ((𝑛𝐴𝑋) supp 𝑍) = ∅)
3533, 34nsyl5 159 . . . 4 𝑍 ∈ V → ((𝑛𝐴𝑋) supp 𝑍) = ∅)
36 simpr 484 . . . . 5 (((𝑛𝐵𝑋) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
37 supp0prc 8188 . . . . 5 (¬ ((𝑛𝐵𝑋) ∈ V ∧ 𝑍 ∈ V) → ((𝑛𝐵𝑋) supp 𝑍) = ∅)
3836, 37nsyl5 159 . . . 4 𝑍 ∈ V → ((𝑛𝐵𝑋) supp 𝑍) = ∅)
3935, 38eqtr4d 2780 . . 3 𝑍 ∈ V → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍))
4039a1d 25 . 2 𝑍 ∈ V → (𝜑 → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍)))
4132, 40pm2.61i 182 1 (𝜑 → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍))
Colors of variables: wff setvar class
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
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