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Theorem extmptsuppeq 8211
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 3993 . . . . . . 7 ((𝑍 ∈ V ∧ 𝜑) → (𝑛𝐴𝑛𝐵))
43anim1d 611 . . . . . 6 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋 ∈ (V ∖ {𝑍})) → (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))))
5 eldif 3972 . . . . . . . . . . . . 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 4668 . . . . . . . . . . . . 13 (𝑍 ∈ V → (𝑋 ∈ {𝑍} ↔ 𝑋 = 𝑍))
11 elndif 4142 . . . . . . . . . . . . 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 3434 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → {𝑛𝐴𝑋 ∈ (V ∖ {𝑍})} = {𝑛𝐵𝑋 ∈ (V ∖ {𝑍})})
22 eqid 2734 . . . . 5 (𝑛𝐴𝑋) = (𝑛𝐴𝑋)
23 extmptsuppeq.b . . . . . . 7 (𝜑𝐵𝑊)
2423, 1ssexd 5329 . . . . . 6 (𝜑𝐴 ∈ V)
2524adantl 481 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → 𝐴 ∈ V)
26 simpl 482 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
2722, 25, 26mptsuppdifd 8209 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋) supp 𝑍) = {𝑛𝐴𝑋 ∈ (V ∖ {𝑍})})
28 eqid 2734 . . . . 5 (𝑛𝐵𝑋) = (𝑛𝐵𝑋)
2923adantl 481 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → 𝐵𝑊)
3028, 29, 26mptsuppdifd 8209 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐵𝑋) supp 𝑍) = {𝑛𝐵𝑋 ∈ (V ∖ {𝑍})})
3121, 27, 303eqtr4d 2784 . . 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 8186 . . . . 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 8186 . . . . 5 (¬ ((𝑛𝐵𝑋) ∈ V ∧ 𝑍 ∈ V) → ((𝑛𝐵𝑋) supp 𝑍) = ∅)
3836, 37nsyl5 159 . . . 4 𝑍 ∈ V → ((𝑛𝐵𝑋) supp 𝑍) = ∅)
3935, 38eqtr4d 2777 . . 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 1536  wcel 2105  {crab 3432  Vcvv 3477  cdif 3959  wss 3962  c0 4338  {csn 4630  cmpt 5230  (class class class)co 7430   supp csupp 8183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-supp 8184
This theorem is referenced by:  cantnfrescl  9713  cantnfres  9714
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