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Theorem ressuppssdif 7485
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 3725 . . . . . 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 4331 . . . . . . . . . 10 (𝑧 = 𝑥 → {𝑧} = {𝑥})
32imaeq2d 5624 . . . . . . . . 9 (𝑧 = 𝑥 → (𝐹 “ {𝑧}) = (𝐹 “ {𝑥}))
43neeq1d 2991 . . . . . . . 8 (𝑧 = 𝑥 → ((𝐹 “ {𝑧}) ≠ {𝑍} ↔ (𝐹 “ {𝑥}) ≠ {𝑍}))
54elrab 3504 . . . . . . 7 (𝑥 ∈ {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}))
6 ianor 510 . . . . . . . 8 (¬ (𝑥 ∈ dom (𝐹𝐵) ∧ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}) ↔ (¬ 𝑥 ∈ dom (𝐹𝐵) ∨ ¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
72imaeq2d 5624 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝐹𝐵) “ {𝑧}) = ((𝐹𝐵) “ {𝑥}))
87neeq1d 2991 . . . . . . . . 9 (𝑧 = 𝑥 → (((𝐹𝐵) “ {𝑧}) ≠ {𝑍} ↔ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
98elrab 3504 . . . . . . . 8 (𝑥 ∈ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}} ↔ (𝑥 ∈ dom (𝐹𝐵) ∧ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
106, 9xchnxbir 322 . . . . . . 7 𝑥 ∈ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}} ↔ (¬ 𝑥 ∈ dom (𝐹𝐵) ∨ ¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
11 ianor 510 . . . . . . . . . . 11 (¬ (𝑥𝐵𝑥 ∈ dom 𝐹) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝑥 ∈ dom 𝐹))
12 dmres 5577 . . . . . . . . . . . 12 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
1312elin2 3944 . . . . . . . . . . 11 (𝑥 ∈ dom (𝐹𝐵) ↔ (𝑥𝐵𝑥 ∈ dom 𝐹))
1411, 13xchnxbir 322 . . . . . . . . . 10 𝑥 ∈ dom (𝐹𝐵) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝑥 ∈ dom 𝐹))
15 simpl 474 . . . . . . . . . . . . . . 15 ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ dom 𝐹)
1615anim2i 594 . . . . . . . . . . . . . 14 ((¬ 𝑥𝐵 ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → (¬ 𝑥𝐵𝑥 ∈ dom 𝐹))
1716ancomd 466 . . . . . . . . . . . . 13 ((¬ 𝑥𝐵 ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥𝐵))
18 eldif 3725 . . . . . . . . . . . . 13 (𝑥 ∈ (dom 𝐹𝐵) ↔ (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥𝐵))
1917, 18sylibr 224 . . . . . . . . . . . 12 ((¬ 𝑥𝐵 ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ (dom 𝐹𝐵))
2019ex 449 . . . . . . . . . . 11 𝑥𝐵 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
21 pm2.24 121 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝐹 → (¬ 𝑥 ∈ dom 𝐹𝑥 ∈ (dom 𝐹𝐵)))
2221adantr 472 . . . . . . . . . . . 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 393 . . . . . . . . . 10 ((¬ 𝑥𝐵 ∨ ¬ 𝑥 ∈ dom 𝐹) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
2514, 24sylbi 207 . . . . . . . . 9 𝑥 ∈ dom (𝐹𝐵) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
2615adantl 473 . . . . . . . . . . 11 ((¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍} ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ dom 𝐹)
27 snssi 4484 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐵 → {𝑥} ⊆ 𝐵)
2827adantl 473 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ dom 𝐹𝑥𝐵) → {𝑥} ⊆ 𝐵)
29 resima2 5590 . . . . . . . . . . . . . . . . . . . 20 ({𝑥} ⊆ 𝐵 → ((𝐹𝐵) “ {𝑥}) = (𝐹 “ {𝑥}))
3028, 29syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ dom 𝐹𝑥𝐵) → ((𝐹𝐵) “ {𝑥}) = (𝐹 “ {𝑥}))
3130eqcomd 2766 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ dom 𝐹𝑥𝐵) → (𝐹 “ {𝑥}) = ((𝐹𝐵) “ {𝑥}))
3231adantr 472 . . . . . . . . . . . . . . . . 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 2794 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵) “ {𝑥}) = {𝑍}) → (𝐹 “ {𝑥}) = {𝑍})
3534ex 449 . . . . . . . . . . . . . . 15 ((𝑥 ∈ dom 𝐹𝑥𝐵) → (((𝐹𝐵) “ {𝑥}) = {𝑍} → (𝐹 “ {𝑥}) = {𝑍}))
3635necon3d 2953 . . . . . . . . . . . . . 14 ((𝑥 ∈ dom 𝐹𝑥𝐵) → ((𝐹 “ {𝑥}) ≠ {𝑍} → ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
3736impancom 455 . . . . . . . . . . . . 13 ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → (𝑥𝐵 → ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}))
3837con3d 148 . . . . . . . . . . . 12 ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → (¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍} → ¬ 𝑥𝐵))
3938impcom 445 . . . . . . . . . . 11 ((¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍} ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → ¬ 𝑥𝐵)
4026, 39eldifd 3726 . . . . . . . . . 10 ((¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍} ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ (dom 𝐹𝐵))
4140ex 449 . . . . . . . . 9 (¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍} → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
4225, 41jaoi 393 . . . . . . . 8 ((¬ 𝑥 ∈ dom (𝐹𝐵) ∨ ¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍}) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹𝐵)))
4342impcom 445 . . . . . . 7 (((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) ∧ (¬ 𝑥 ∈ dom (𝐹𝐵) ∨ ¬ ((𝐹𝐵) “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ (dom 𝐹𝐵))
445, 10, 43syl2anb 497 . . . . . 6 ((𝑥 ∈ {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∧ ¬ 𝑥 ∈ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}}) → 𝑥 ∈ (dom 𝐹𝐵))
451, 44sylbi 207 . . . . 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 3750 . . 3 ((𝐹𝑉𝑍𝑊) → ({𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∖ {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}}) ⊆ (dom 𝐹𝐵))
48 ssundif 4196 . . 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 224 . 2 ((𝐹𝑉𝑍𝑊) → {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ⊆ ({𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}} ∪ (dom 𝐹𝐵)))
50 suppval 7466 . 2 ((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}})
51 resexg 5600 . . . 4 (𝐹𝑉 → (𝐹𝐵) ∈ V)
52 suppval 7466 . . . 4 (((𝐹𝐵) ∈ V ∧ 𝑍𝑊) → ((𝐹𝐵) supp 𝑍) = {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}})
5351, 52sylan 489 . . 3 ((𝐹𝑉𝑍𝑊) → ((𝐹𝐵) supp 𝑍) = {𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}})
5453uneq1d 3909 . 2 ((𝐹𝑉𝑍𝑊) → (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)) = ({𝑧 ∈ dom (𝐹𝐵) ∣ ((𝐹𝐵) “ {𝑧}) ≠ {𝑍}} ∪ (dom 𝐹𝐵)))
5549, 50, 543sstr4d 3789 1 ((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) ⊆ (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1632  wcel 2139  wne 2932  {crab 3054  Vcvv 3340  cdif 3712  cun 3713  wss 3715  {csn 4321  dom cdm 5266  cres 5268  cima 5269  (class class class)co 6814   supp csupp 7464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-supp 7465
This theorem is referenced by:  ressuppfi  8468
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