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Theorem fdifsuppconst 30452
Description: A function is a zero constant outside of its support. (Contributed by Thierry Arnoux, 22-Jun-2024.)
Hypothesis
Ref Expression
fdifsuppconst.1 𝐴 = (dom 𝐹 ∖ (𝐹 supp 𝑍))
Assertion
Ref Expression
fdifsuppconst ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹𝐴) = (𝐴 × {𝑍}))

Proof of Theorem fdifsuppconst
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 funfn 6358 . . . . . 6 (Fun 𝐹𝐹 Fn dom 𝐹)
21biimpi 219 . . . . 5 (Fun 𝐹𝐹 Fn dom 𝐹)
32ad2antrr 725 . . . 4 (((Fun 𝐹𝐹𝑉) ∧ 𝑍𝑊) → 𝐹 Fn dom 𝐹)
4 fdifsuppconst.1 . . . . 5 𝐴 = (dom 𝐹 ∖ (𝐹 supp 𝑍))
5 difssd 4063 . . . . 5 (((Fun 𝐹𝐹𝑉) ∧ 𝑍𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) ⊆ dom 𝐹)
64, 5eqsstrid 3966 . . . 4 (((Fun 𝐹𝐹𝑉) ∧ 𝑍𝑊) → 𝐴 ⊆ dom 𝐹)
73, 6fnssresd 6447 . . 3 (((Fun 𝐹𝐹𝑉) ∧ 𝑍𝑊) → (𝐹𝐴) Fn 𝐴)
8 fnconstg 6545 . . . 4 (𝑍𝑊 → (𝐴 × {𝑍}) Fn 𝐴)
98adantl 485 . . 3 (((Fun 𝐹𝐹𝑉) ∧ 𝑍𝑊) → (𝐴 × {𝑍}) Fn 𝐴)
103adantr 484 . . . . 5 ((((Fun 𝐹𝐹𝑉) ∧ 𝑍𝑊) ∧ 𝑥𝐴) → 𝐹 Fn dom 𝐹)
11 dmexg 7598 . . . . . 6 (𝐹𝑉 → dom 𝐹 ∈ V)
1211ad3antlr 730 . . . . 5 ((((Fun 𝐹𝐹𝑉) ∧ 𝑍𝑊) ∧ 𝑥𝐴) → dom 𝐹 ∈ V)
13 simplr 768 . . . . 5 ((((Fun 𝐹𝐹𝑉) ∧ 𝑍𝑊) ∧ 𝑥𝐴) → 𝑍𝑊)
144eleq2i 2884 . . . . . . 7 (𝑥𝐴𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)))
1514biimpi 219 . . . . . 6 (𝑥𝐴𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)))
1615adantl 485 . . . . 5 ((((Fun 𝐹𝐹𝑉) ∧ 𝑍𝑊) ∧ 𝑥𝐴) → 𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)))
1710, 12, 13, 16fvdifsupp 30447 . . . 4 ((((Fun 𝐹𝐹𝑉) ∧ 𝑍𝑊) ∧ 𝑥𝐴) → (𝐹𝑥) = 𝑍)
18 simpr 488 . . . . 5 ((((Fun 𝐹𝐹𝑉) ∧ 𝑍𝑊) ∧ 𝑥𝐴) → 𝑥𝐴)
1918fvresd 6669 . . . 4 ((((Fun 𝐹𝐹𝑉) ∧ 𝑍𝑊) ∧ 𝑥𝐴) → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
20 fvconst2g 6945 . . . . 5 ((𝑍𝑊𝑥𝐴) → ((𝐴 × {𝑍})‘𝑥) = 𝑍)
2120adantll 713 . . . 4 ((((Fun 𝐹𝐹𝑉) ∧ 𝑍𝑊) ∧ 𝑥𝐴) → ((𝐴 × {𝑍})‘𝑥) = 𝑍)
2217, 19, 213eqtr4d 2846 . . 3 ((((Fun 𝐹𝐹𝑉) ∧ 𝑍𝑊) ∧ 𝑥𝐴) → ((𝐹𝐴)‘𝑥) = ((𝐴 × {𝑍})‘𝑥))
237, 9, 22eqfnfvd 6786 . 2 (((Fun 𝐹𝐹𝑉) ∧ 𝑍𝑊) → (𝐹𝐴) = (𝐴 × {𝑍}))
24233impa 1107 1 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹𝐴) = (𝐴 × {𝑍}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  Vcvv 3444  cdif 3881  {csn 4528   × cxp 5521  dom cdm 5523  cres 5525  Fun wfun 6322   Fn wfn 6323  cfv 6328  (class class class)co 7139   supp csupp 7817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-supp 7818
This theorem is referenced by: (None)
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