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Theorem mptsuppdifd 7586
Description: The support of a function in maps-to notation with a class difference. (Contributed by AV, 28-May-2019.)
Hypotheses
Ref Expression
mptsuppdifd.f 𝐹 = (𝑥𝐴𝐵)
mptsuppdifd.a (𝜑𝐴𝑉)
mptsuppdifd.z (𝜑𝑍𝑊)
Assertion
Ref Expression
mptsuppdifd (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})})
Distinct variable groups:   𝑥,𝐴   𝑥,𝑍
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem mptsuppdifd
StepHypRef Expression
1 mptsuppdifd.f . . . 4 𝐹 = (𝑥𝐴𝐵)
2 mptsuppdifd.a . . . . 5 (𝜑𝐴𝑉)
3 mptexg 6745 . . . . 5 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
42, 3syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐵) ∈ V)
51, 4syl5eqel 2910 . . 3 (𝜑𝐹 ∈ V)
6 mptsuppdifd.z . . 3 (𝜑𝑍𝑊)
7 suppimacnv 7575 . . 3 ((𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
85, 6, 7syl2anc 579 . 2 (𝜑 → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
91mptpreima 5873 . 2 (𝐹 “ (V ∖ {𝑍})) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})}
108, 9syl6eq 2877 1 (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  {crab 3121  Vcvv 3414  cdif 3795  {csn 4399  cmpt 4954  ccnv 5345  cima 5349  (class class class)co 6910   supp csupp 7564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-supp 7565
This theorem is referenced by:  mptsuppd  7587  extmptsuppeq  7588  suppssov1  7597  suppss2  7599  suppssfv  7601
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