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Theorem suppfnss 7280
Description: The support of a function which has the same zero values (in its domain) as another function is a subset of the support of this other function. (Contributed by AV, 30-Apr-2019.)
Assertion
Ref Expression
suppfnss (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝑥,𝑍
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem suppfnss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fndm 5958 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
21eleq2d 2684 . . . . . . . . . 10 (𝐹 Fn 𝐴 → (𝑦 ∈ dom 𝐹𝑦𝐴))
32ad2antrr 761 . . . . . . . . 9 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝑦 ∈ dom 𝐹𝑦𝐴))
4 fveq2 6158 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝐺𝑥) = (𝐺𝑦))
54eqeq1d 2623 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝐺𝑥) = 𝑍 ↔ (𝐺𝑦) = 𝑍))
6 fveq2 6158 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
76eqeq1d 2623 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝐹𝑥) = 𝑍 ↔ (𝐹𝑦) = 𝑍))
85, 7imbi12d 334 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) ↔ ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍)))
98rspcv 3295 . . . . . . . . 9 (𝑦𝐴 → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍)))
103, 9syl6bi 243 . . . . . . . 8 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝑦 ∈ dom 𝐹 → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍))))
1110com23 86 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → (𝑦 ∈ dom 𝐹 → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍))))
1211imp31 448 . . . . . 6 (((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍))
1312necon3d 2811 . . . . 5 (((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐹𝑦) ≠ 𝑍 → (𝐺𝑦) ≠ 𝑍))
1413ss2rabdv 3668 . . . 4 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐹 ∣ (𝐺𝑦) ≠ 𝑍})
15 simpr1 1065 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝐴𝐵)
161ad2antrr 761 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → dom 𝐹 = 𝐴)
17 fndm 5958 . . . . . . . 8 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
1817ad2antlr 762 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → dom 𝐺 = 𝐵)
1915, 16, 183sstr4d 3633 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → dom 𝐹 ⊆ dom 𝐺)
2019adantr 481 . . . . 5 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → dom 𝐹 ⊆ dom 𝐺)
21 rabss2 3670 . . . . 5 (dom 𝐹 ⊆ dom 𝐺 → {𝑦 ∈ dom 𝐹 ∣ (𝐺𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
2220, 21syl 17 . . . 4 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → {𝑦 ∈ dom 𝐹 ∣ (𝐺𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
2314, 22sstrd 3598 . . 3 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
24 fnfun 5956 . . . . . . 7 (𝐹 Fn 𝐴 → Fun 𝐹)
2524ad2antrr 761 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → Fun 𝐹)
26 simpl 473 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐹 Fn 𝐴)
27 ssexg 4774 . . . . . . . 8 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
28273adant3 1079 . . . . . . 7 ((𝐴𝐵𝐵𝑉𝑍𝑊) → 𝐴 ∈ V)
29 fnex 6446 . . . . . . 7 ((𝐹 Fn 𝐴𝐴 ∈ V) → 𝐹 ∈ V)
3026, 28, 29syl2an 494 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝐹 ∈ V)
31 simpr3 1067 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝑍𝑊)
32 suppval1 7261 . . . . . 6 ((Fun 𝐹𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍})
3325, 30, 31, 32syl3anc 1323 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝐹 supp 𝑍) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍})
34 fnfun 5956 . . . . . . 7 (𝐺 Fn 𝐵 → Fun 𝐺)
3534ad2antlr 762 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → Fun 𝐺)
36 simpr 477 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐺 Fn 𝐵)
37 simp2 1060 . . . . . . 7 ((𝐴𝐵𝐵𝑉𝑍𝑊) → 𝐵𝑉)
38 fnex 6446 . . . . . . 7 ((𝐺 Fn 𝐵𝐵𝑉) → 𝐺 ∈ V)
3936, 37, 38syl2an 494 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝐺 ∈ V)
40 suppval1 7261 . . . . . 6 ((Fun 𝐺𝐺 ∈ V ∧ 𝑍𝑊) → (𝐺 supp 𝑍) = {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
4135, 39, 31, 40syl3anc 1323 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝐺 supp 𝑍) = {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
4233, 41sseq12d 3619 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → ((𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍) ↔ {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍}))
4342adantr 481 . . 3 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → ((𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍) ↔ {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍}))
4423, 43mpbird 247 . 2 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
4544ex 450 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2908  {crab 2912  Vcvv 3190  wss 3560  dom cdm 5084  Fun wfun 5851   Fn wfn 5852  cfv 5857  (class class class)co 6615   supp csupp 7255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-supp 7256
This theorem is referenced by:  funsssuppss  7281  suppofss1d  7292  suppofss2d  7293  lincresunit2  41585
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