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Theorem extmptsuppeq 7953
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 485 . . . . . . . 8 ((𝑍 ∈ V ∧ 𝜑) → 𝐴𝐵)
32sseld 3916 . . . . . . 7 ((𝑍 ∈ V ∧ 𝜑) → (𝑛𝐴𝑛𝐵))
43anim1d 614 . . . . . 6 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋 ∈ (V ∖ {𝑍})) → (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))))
5 eldif 3893 . . . . . . . . . . . . 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 598 . . . . . . . . . . . 12 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵 ∧ ¬ 𝑛𝐴)) → 𝑋 = 𝑍)
98expr 460 . . . . . . . . . . 11 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛𝐵) → (¬ 𝑛𝐴𝑋 = 𝑍))
10 elsn2g 4595 . . . . . . . . . . . . 13 (𝑍 ∈ V → (𝑋 ∈ {𝑍} ↔ 𝑋 = 𝑍))
11 elndif 4059 . . . . . . . . . . . . 13 (𝑋 ∈ {𝑍} → ¬ 𝑋 ∈ (V ∖ {𝑍}))
1210, 11syl6bir 257 . . . . . . . . . . . 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 458 . . . . . . . 8 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))) → 𝑛𝐴)
17 simprr 773 . . . . . . . 8 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))) → 𝑋 ∈ (V ∖ {𝑍}))
1816, 17jca 515 . . . . . . 7 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))) → (𝑛𝐴𝑋 ∈ (V ∖ {𝑍})))
1918ex 416 . . . . . 6 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐵𝑋 ∈ (V ∖ {𝑍})) → (𝑛𝐴𝑋 ∈ (V ∖ {𝑍}))))
204, 19impbid 215 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋 ∈ (V ∖ {𝑍})) ↔ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))))
2120rabbidva2 3401 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → {𝑛𝐴𝑋 ∈ (V ∖ {𝑍})} = {𝑛𝐵𝑋 ∈ (V ∖ {𝑍})})
22 eqid 2739 . . . . 5 (𝑛𝐴𝑋) = (𝑛𝐴𝑋)
23 extmptsuppeq.b . . . . . . 7 (𝜑𝐵𝑊)
2423, 1ssexd 5233 . . . . . 6 (𝜑𝐴 ∈ V)
2524adantl 485 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → 𝐴 ∈ V)
26 simpl 486 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
2722, 25, 26mptsuppdifd 7951 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋) supp 𝑍) = {𝑛𝐴𝑋 ∈ (V ∖ {𝑍})})
28 eqid 2739 . . . . 5 (𝑛𝐵𝑋) = (𝑛𝐵𝑋)
2923adantl 485 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → 𝐵𝑊)
3028, 29, 26mptsuppdifd 7951 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐵𝑋) supp 𝑍) = {𝑛𝐵𝑋 ∈ (V ∖ {𝑍})})
3121, 27, 303eqtr4d 2789 . . 3 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍))
3231ex 416 . 2 (𝑍 ∈ V → (𝜑 → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍)))
33 simpr 488 . . . . 5 (((𝑛𝐴𝑋) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
34 supp0prc 7929 . . . . 5 (¬ ((𝑛𝐴𝑋) ∈ V ∧ 𝑍 ∈ V) → ((𝑛𝐴𝑋) supp 𝑍) = ∅)
3533, 34nsyl5 162 . . . 4 𝑍 ∈ V → ((𝑛𝐴𝑋) supp 𝑍) = ∅)
36 simpr 488 . . . . 5 (((𝑛𝐵𝑋) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
37 supp0prc 7929 . . . . 5 (¬ ((𝑛𝐵𝑋) ∈ V ∧ 𝑍 ∈ V) → ((𝑛𝐵𝑋) supp 𝑍) = ∅)
3836, 37nsyl5 162 . . . 4 𝑍 ∈ V → ((𝑛𝐵𝑋) supp 𝑍) = ∅)
3935, 38eqtr4d 2782 . . 3 𝑍 ∈ V → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍))
4039a1d 25 . 2 𝑍 ∈ V → (𝜑 → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍)))
4132, 40pm2.61i 185 1 (𝜑 → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1543  wcel 2112  {crab 3068  Vcvv 3423  cdif 3880  wss 3883  c0 4253  {csn 4557  cmpt 5151  (class class class)co 7234   supp csupp 7926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5195  ax-sep 5208  ax-nul 5215  ax-pr 5338  ax-un 7544
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-iun 4922  df-br 5070  df-opab 5132  df-mpt 5152  df-id 5471  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408  df-ov 7237  df-oprab 7238  df-mpo 7239  df-supp 7927
This theorem is referenced by:  cantnfrescl  9320  cantnfres  9321
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