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Theorem ressuppssdif 7580
 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 3808 . . . . . 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 4407 . . . . . . . . . 10 (𝑧 = 𝑥 → {𝑧} = {𝑥})
32imaeq2d 5707 . . . . . . . . 9 (𝑧 = 𝑥 → (𝐹 “ {𝑧}) = (𝐹 “ {𝑥}))
43neeq1d 3058 . . . . . . . 8 (𝑧 = 𝑥 → ((𝐹 “ {𝑧}) ≠ {𝑍} ↔ (𝐹 “ {𝑥}) ≠ {𝑍}))
54elrab 3585 . . . . . . 7 (𝑥 ∈ {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}))
6 ianor 1011 . . . . . . . 8 (¬ (𝑥 ∈ dom (𝐹𝐵) ∧ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}) ↔ (¬ 𝑥 ∈ dom (𝐹𝐵) ∨ ¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
72imaeq2d 5707 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝐹𝐵) “ {𝑧}) = ((𝐹𝐵) “ {𝑥}))
87neeq1d 3058 . . . . . . . . 9 (𝑧 = 𝑥 → (((𝐹𝐵) “ {𝑧}) ≠ {𝑍} ↔ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
98elrab 3585 . . . . . . . 8 (𝑥 ∈ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}} ↔ (𝑥 ∈ dom (𝐹𝐵) ∧ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
106, 9xchnxbir 325 . . . . . . 7 𝑥 ∈ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}} ↔ (¬ 𝑥 ∈ dom (𝐹𝐵) ∨ ¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
11 ianor 1011 . . . . . . . . . . 11 (¬ (𝑥𝐵𝑥 ∈ dom 𝐹) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝑥 ∈ dom 𝐹))
12 dmres 5655 . . . . . . . . . . . 12 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
1312elin2 4028 . . . . . . . . . . 11 (𝑥 ∈ dom (𝐹𝐵) ↔ (𝑥𝐵𝑥 ∈ dom 𝐹))
1411, 13xchnxbir 325 . . . . . . . . . 10 𝑥 ∈ dom (𝐹𝐵) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝑥 ∈ dom 𝐹))
15 simpl 476 . . . . . . . . . . . . . . 15 ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ dom 𝐹)
1615anim2i 612 . . . . . . . . . . . . . 14 ((¬ 𝑥𝐵 ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → (¬ 𝑥𝐵𝑥 ∈ dom 𝐹))
1716ancomd 455 . . . . . . . . . . . . 13 ((¬ 𝑥𝐵 ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥𝐵))
18 eldif 3808 . . . . . . . . . . . . 13 (𝑥 ∈ (dom 𝐹𝐵) ↔ (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥𝐵))
1917, 18sylibr 226 . . . . . . . . . . . 12 ((¬ 𝑥𝐵 ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ (dom 𝐹𝐵))
2019ex 403 . . . . . . . . . . 11 𝑥𝐵 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
21 pm2.24 122 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝐹 → (¬ 𝑥 ∈ dom 𝐹𝑥 ∈ (dom 𝐹𝐵)))
2221adantr 474 . . . . . . . . . . . 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 890 . . . . . . . . . 10 ((¬ 𝑥𝐵 ∨ ¬ 𝑥 ∈ dom 𝐹) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
2514, 24sylbi 209 . . . . . . . . 9 𝑥 ∈ dom (𝐹𝐵) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
2615adantl 475 . . . . . . . . . . 11 ((¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍} ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ dom 𝐹)
27 snssi 4557 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐵 → {𝑥} ⊆ 𝐵)
2827adantl 475 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ dom 𝐹𝑥𝐵) → {𝑥} ⊆ 𝐵)
29 resima2 5668 . . . . . . . . . . . . . . . . . . . 20 ({𝑥} ⊆ 𝐵 → ((𝐹𝐵) “ {𝑥}) = (𝐹 “ {𝑥}))
3028, 29syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ dom 𝐹𝑥𝐵) → ((𝐹𝐵) “ {𝑥}) = (𝐹 “ {𝑥}))
3130eqcomd 2831 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ dom 𝐹𝑥𝐵) → (𝐹 “ {𝑥}) = ((𝐹𝐵) “ {𝑥}))
3231adantr 474 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵) “ {𝑥}) = {𝑍}) → (𝐹 “ {𝑥}) = ((𝐹𝐵) “ {𝑥}))
33 simpr 479 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵) “ {𝑥}) = {𝑍}) → ((𝐹𝐵) “ {𝑥}) = {𝑍})
3432, 33eqtrd 2861 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵) “ {𝑥}) = {𝑍}) → (𝐹 “ {𝑥}) = {𝑍})
3534ex 403 . . . . . . . . . . . . . . 15 ((𝑥 ∈ dom 𝐹𝑥𝐵) → (((𝐹𝐵) “ {𝑥}) = {𝑍} → (𝐹 “ {𝑥}) = {𝑍}))
3635necon3d 3020 . . . . . . . . . . . . . 14 ((𝑥 ∈ dom 𝐹𝑥𝐵) → ((𝐹 “ {𝑥}) ≠ {𝑍} → ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
3736impancom 445 . . . . . . . . . . . . 13 ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → (𝑥𝐵 → ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
3837con3d 150 . . . . . . . . . . . 12 ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → (¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍} → ¬ 𝑥𝐵))
3938impcom 398 . . . . . . . . . . 11 ((¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍} ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → ¬ 𝑥𝐵)
4026, 39eldifd 3809 . . . . . . . . . 10 ((¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍} ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ (dom 𝐹𝐵))
4140ex 403 . . . . . . . . 9 (¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍} → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
4225, 41jaoi 890 . . . . . . . 8 ((¬ 𝑥 ∈ dom (𝐹𝐵) ∨ ¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
4342impcom 398 . . . . . . 7 (((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) ∧ (¬ 𝑥 ∈ dom (𝐹𝐵) ∨ ¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ (dom 𝐹𝐵))
445, 10, 43syl2anb 593 . . . . . 6 ((𝑥 ∈ {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∧ ¬ 𝑥 ∈ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}}) → 𝑥 ∈ (dom 𝐹𝐵))
451, 44sylbi 209 . . . . 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 3833 . . 3 ((𝐹𝑉𝑍𝑊) → ({𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∖ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}}) ⊆ (dom 𝐹𝐵))
48 ssundif 4275 . . 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 226 . 2 ((𝐹𝑉𝑍𝑊) → {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ⊆ ({𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}} ∪ (dom 𝐹𝐵)))
50 suppval 7561 . 2 ((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}})
51 resexg 5679 . . . 4 (𝐹𝑉 → (𝐹𝐵) ∈ V)
52 suppval 7561 . . . 4 (((𝐹𝐵) ∈ V ∧ 𝑍𝑊) → ((𝐹𝐵) supp 𝑍) = {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}})
5351, 52sylan 577 . . 3 ((𝐹𝑉𝑍𝑊) → ((𝐹𝐵) supp 𝑍) = {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}})
5453uneq1d 3993 . 2 ((𝐹𝑉𝑍𝑊) → (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)) = ({𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}} ∪ (dom 𝐹𝐵)))
5549, 50, 543sstr4d 3873 1 ((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) ⊆ (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   ∨ wo 880   = wceq 1658   ∈ wcel 2166   ≠ wne 2999  {crab 3121  Vcvv 3414   ∖ cdif 3795   ∪ cun 3796   ⊆ wss 3798  {csn 4397  dom cdm 5342   ↾ cres 5344   “ cima 5345  (class class class)co 6905   supp csupp 7559 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  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-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-supp 7560 This theorem is referenced by:  ressuppfi  8570
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