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Theorem ressuppssdif 7840
Description: The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)
Assertion
Ref Expression
ressuppssdif ((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) ⊆ (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)))

Proof of Theorem ressuppssdif
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldif 3943 . . . . . 6 (𝑥 ∈ ({𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∖ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}}) ↔ (𝑥 ∈ {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∧ ¬ 𝑥 ∈ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}}))
2 sneq 4567 . . . . . . . . . 10 (𝑧 = 𝑥 → {𝑧} = {𝑥})
32imaeq2d 5922 . . . . . . . . 9 (𝑧 = 𝑥 → (𝐹 “ {𝑧}) = (𝐹 “ {𝑥}))
43neeq1d 3072 . . . . . . . 8 (𝑧 = 𝑥 → ((𝐹 “ {𝑧}) ≠ {𝑍} ↔ (𝐹 “ {𝑥}) ≠ {𝑍}))
54elrab 3677 . . . . . . 7 (𝑥 ∈ {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}))
6 ianor 975 . . . . . . . 8 (¬ (𝑥 ∈ dom (𝐹𝐵) ∧ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}) ↔ (¬ 𝑥 ∈ dom (𝐹𝐵) ∨ ¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
72imaeq2d 5922 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝐹𝐵) “ {𝑧}) = ((𝐹𝐵) “ {𝑥}))
87neeq1d 3072 . . . . . . . . 9 (𝑧 = 𝑥 → (((𝐹𝐵) “ {𝑧}) ≠ {𝑍} ↔ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
98elrab 3677 . . . . . . . 8 (𝑥 ∈ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}} ↔ (𝑥 ∈ dom (𝐹𝐵) ∧ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
106, 9xchnxbir 334 . . . . . . 7 𝑥 ∈ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}} ↔ (¬ 𝑥 ∈ dom (𝐹𝐵) ∨ ¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
11 ianor 975 . . . . . . . . . . 11 (¬ (𝑥𝐵𝑥 ∈ dom 𝐹) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝑥 ∈ dom 𝐹))
12 dmres 5868 . . . . . . . . . . . 12 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
1312elin2 4171 . . . . . . . . . . 11 (𝑥 ∈ dom (𝐹𝐵) ↔ (𝑥𝐵𝑥 ∈ dom 𝐹))
1411, 13xchnxbir 334 . . . . . . . . . 10 𝑥 ∈ dom (𝐹𝐵) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝑥 ∈ dom 𝐹))
15 simpl 483 . . . . . . . . . . . . . . 15 ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ dom 𝐹)
1615anim2i 616 . . . . . . . . . . . . . 14 ((¬ 𝑥𝐵 ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → (¬ 𝑥𝐵𝑥 ∈ dom 𝐹))
1716ancomd 462 . . . . . . . . . . . . 13 ((¬ 𝑥𝐵 ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥𝐵))
18 eldif 3943 . . . . . . . . . . . . 13 (𝑥 ∈ (dom 𝐹𝐵) ↔ (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥𝐵))
1917, 18sylibr 235 . . . . . . . . . . . 12 ((¬ 𝑥𝐵 ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ (dom 𝐹𝐵))
2019ex 413 . . . . . . . . . . 11 𝑥𝐵 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
21 pm2.24 124 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝐹 → (¬ 𝑥 ∈ dom 𝐹𝑥 ∈ (dom 𝐹𝐵)))
2221adantr 481 . . . . . . . . . . . 12 ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → (¬ 𝑥 ∈ dom 𝐹𝑥 ∈ (dom 𝐹𝐵)))
2322com12 32 . . . . . . . . . . 11 𝑥 ∈ dom 𝐹 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
2420, 23jaoi 851 . . . . . . . . . 10 ((¬ 𝑥𝐵 ∨ ¬ 𝑥 ∈ dom 𝐹) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
2514, 24sylbi 218 . . . . . . . . 9 𝑥 ∈ dom (𝐹𝐵) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
2615adantl 482 . . . . . . . . . . 11 ((¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍} ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ dom 𝐹)
27 snssi 4733 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐵 → {𝑥} ⊆ 𝐵)
2827adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ dom 𝐹𝑥𝐵) → {𝑥} ⊆ 𝐵)
29 resima2 5881 . . . . . . . . . . . . . . . . . . . 20 ({𝑥} ⊆ 𝐵 → ((𝐹𝐵) “ {𝑥}) = (𝐹 “ {𝑥}))
3028, 29syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ dom 𝐹𝑥𝐵) → ((𝐹𝐵) “ {𝑥}) = (𝐹 “ {𝑥}))
3130eqcomd 2824 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ dom 𝐹𝑥𝐵) → (𝐹 “ {𝑥}) = ((𝐹𝐵) “ {𝑥}))
3231adantr 481 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵) “ {𝑥}) = {𝑍}) → (𝐹 “ {𝑥}) = ((𝐹𝐵) “ {𝑥}))
33 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵) “ {𝑥}) = {𝑍}) → ((𝐹𝐵) “ {𝑥}) = {𝑍})
3432, 33eqtrd 2853 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵) “ {𝑥}) = {𝑍}) → (𝐹 “ {𝑥}) = {𝑍})
3534ex 413 . . . . . . . . . . . . . . 15 ((𝑥 ∈ dom 𝐹𝑥𝐵) → (((𝐹𝐵) “ {𝑥}) = {𝑍} → (𝐹 “ {𝑥}) = {𝑍}))
3635necon3d 3034 . . . . . . . . . . . . . 14 ((𝑥 ∈ dom 𝐹𝑥𝐵) → ((𝐹 “ {𝑥}) ≠ {𝑍} → ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
3736impancom 452 . . . . . . . . . . . . 13 ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → (𝑥𝐵 → ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
3837con3d 155 . . . . . . . . . . . 12 ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → (¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍} → ¬ 𝑥𝐵))
3938impcom 408 . . . . . . . . . . 11 ((¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍} ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → ¬ 𝑥𝐵)
4026, 39eldifd 3944 . . . . . . . . . 10 ((¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍} ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ (dom 𝐹𝐵))
4140ex 413 . . . . . . . . 9 (¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍} → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
4225, 41jaoi 851 . . . . . . . 8 ((¬ 𝑥 ∈ dom (𝐹𝐵) ∨ ¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
4342impcom 408 . . . . . . 7 (((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) ∧ (¬ 𝑥 ∈ dom (𝐹𝐵) ∨ ¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ (dom 𝐹𝐵))
445, 10, 43syl2anb 597 . . . . . 6 ((𝑥 ∈ {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∧ ¬ 𝑥 ∈ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}}) → 𝑥 ∈ (dom 𝐹𝐵))
451, 44sylbi 218 . . . . 5 (𝑥 ∈ ({𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∖ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}}) → 𝑥 ∈ (dom 𝐹𝐵))
4645a1i 11 . . . 4 ((𝐹𝑉𝑍𝑊) → (𝑥 ∈ ({𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∖ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}}) → 𝑥 ∈ (dom 𝐹𝐵)))
4746ssrdv 3970 . . 3 ((𝐹𝑉𝑍𝑊) → ({𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∖ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}}) ⊆ (dom 𝐹𝐵))
48 ssundif 4429 . . 3 ({𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ⊆ ({𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}} ∪ (dom 𝐹𝐵)) ↔ ({𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∖ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}}) ⊆ (dom 𝐹𝐵))
4947, 48sylibr 235 . 2 ((𝐹𝑉𝑍𝑊) → {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ⊆ ({𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}} ∪ (dom 𝐹𝐵)))
50 suppval 7821 . 2 ((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}})
51 resexg 5891 . . . 4 (𝐹𝑉 → (𝐹𝐵) ∈ V)
52 suppval 7821 . . . 4 (((𝐹𝐵) ∈ V ∧ 𝑍𝑊) → ((𝐹𝐵) supp 𝑍) = {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}})
5351, 52sylan 580 . . 3 ((𝐹𝑉𝑍𝑊) → ((𝐹𝐵) supp 𝑍) = {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}})
5453uneq1d 4135 . 2 ((𝐹𝑉𝑍𝑊) → (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)) = ({𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}} ∪ (dom 𝐹𝐵)))
5549, 50, 543sstr4d 4011 1 ((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) ⊆ (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  wne 3013  {crab 3139  Vcvv 3492  cdif 3930  cun 3931  wss 3933  {csn 4557  dom cdm 5548  cres 5550  cima 5551  (class class class)co 7145   supp csupp 7819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-supp 7820
This theorem is referenced by:  ressuppfi  8847
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