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Theorem suppssr 8074

Description: A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)

**Hypotheses**
Ref	Expression
suppssr.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
suppssr.n	⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
suppssr.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
suppssr.z	⊢ (𝜑 → 𝑍 ∈ 𝑈)

**Assertion**
Ref	Expression
suppssr	⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍)

**Proof of Theorem suppssr**
Step	Hyp	Ref	Expression
1		eldif 3907	. 2 ⊢ (𝑋 ∈ (𝐴 ∖ 𝑊) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊))
2		fvex 6832	. . . . . 6 ⊢ (𝐹‘𝑋) ∈ V
3		eldifsn 4733	. . . . . 6 ⊢ ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) ↔ ((𝐹‘𝑋) ∈ V ∧ (𝐹‘𝑋) ≠ 𝑍))
4	2, 3	mpbiran 706	. . . . 5 ⊢ ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) ↔ (𝐹‘𝑋) ≠ 𝑍)
5		suppssr.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
6	5	ffnd 6646	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝐴)
7		suppssr.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8		suppssr.z	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
9		elsuppfn 8049	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
10	6, 7, 8, 9	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
11		ibar 529	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑋) ∈ V → ((𝐹‘𝑋) ≠ 𝑍 ↔ ((𝐹‘𝑋) ∈ V ∧ (𝐹‘𝑋) ≠ 𝑍)))
12	2, 11	mp1i 13	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 ↔ ((𝐹‘𝑋) ∈ V ∧ (𝐹‘𝑋) ≠ 𝑍)))
13	12, 3	bitr4di 288	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 ↔ (𝐹‘𝑋) ∈ (V ∖ {𝑍})))
14	13	pm5.32da 579	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍}))))
15	10, 14	bitrd 278	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍}))))
16		suppssr.n	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
17	16	sseld 3930	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) → 𝑋 ∈ 𝑊))
18	15, 17	sylbird 259	. . . . . 6 ⊢ (𝜑 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍})) → 𝑋 ∈ 𝑊))
19	18	expdimp 453	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) → 𝑋 ∈ 𝑊))
20	4, 19	syl5bir 242	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 → 𝑋 ∈ 𝑊))
21	20	necon1bd 2958	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (¬ 𝑋 ∈ 𝑊 → (𝐹‘𝑋) = 𝑍))
22	21	impr 455	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊)) → (𝐹‘𝑋) = 𝑍)
23	1, 22	sylan2b 594	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 Vcvv 3441 ∖ cdif 3894 ⊆ wss 3897 {csn 4572 Fn wfn 6468 ⟶wf 6469 ‘cfv 6473 (class class class)co 7329 supp csupp 8039

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pr 5369 ax-un 7642

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4269 df-if 4473 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-id 5512 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-ov 7332 df-oprab 7333 df-mpo 7334 df-supp 8040

This theorem is referenced by: fsuppmptif 9248 fsuppco2 9252 fsuppcor 9253 cantnfp1lem1 9527 cantnfp1lem3 9529 cantnflem1 9538 cnfcom2lem 9550 gsumval3 19595 gsumcllem 19596 gsumzaddlem 19609 gsumzmhm 19625 gsumpt 19650 gsum2dlem1 19658 gsum2dlem2 19659 gsum2d 19660 gsumxp2 19668 dprdfinv 19709 dprdfadd 19710 dmdprdsplitlem 19727 dpjidcl 19748 gsumdixp 19935 lcomfsupp 20261 uvcresum 21098 frlmsslsp 21101 psrbaglesuppOLD 21226 psrbagaddclOLD 21230 psrbaglefiOLD 21234 mplsubglem 21303 mpllsslem 21304 mplsubrglem 21308 mplmonmul 21335 mplcoe1 21336 mplcoe5 21339 mplbas2 21341 evlslem4 21382 evlslem2 21387 mhpmulcl 21437 mhpvscacl 21442 rrxcph 24654 rrxmval 24667 rrxmetlem 24669 rrxmet 24670 rrxdstprj1 24671 deg1mul3le 25379 suppovss 31217 elrspunidl 31816 fedgmullem1 31921 eulerpartlemb 32548 evlsbagval 40525 fsuppssindlem1 40530

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