Theorem suppss3 32638

Description: Deduce a function's support's inclusion in another function's support. (Contributed by Thierry Arnoux, 7-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)

Hypotheses

Ref	Expression
suppss3.1	⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐵)
suppss3.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
suppss3.z	⊢ (𝜑 → 𝑍 ∈ 𝑊)
suppss3.2	⊢ (𝜑 → 𝐹 Fn 𝐴)
suppss3.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑍) → 𝐵 = 𝑍)

Assertion

Ref	Expression
suppss3	⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝑍   𝜑,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝐺(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem suppss3

Step	Hyp	Ref	Expression
1		suppss3.1	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	oveq1i 7426	. 2 ⊢ (𝐺 supp 𝑍) = ((𝑥 ∈ 𝐴 ↦ 𝐵) supp 𝑍)
3		simpl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝜑)
4		eldifi 4123	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴)
5	4	adantl 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝑥 ∈ 𝐴)
6		suppss3.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 Fn 𝐴)
7		suppss3.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8		fnex 7226	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
9	6, 7, 8	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
10		suppss3.z	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
11		suppimacnv 8180	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
12	9, 10, 11	syl2anc 582	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
13	12	eleq2d 2812	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ 𝑥 ∈ (◡𝐹 “ (V ∖ {𝑍}))))
14		elpreima 7063	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (◡𝐹 “ (V ∖ {𝑍})) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))))
15	6, 14	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (◡𝐹 “ (V ∖ {𝑍})) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))))
16	13, 15	bitrd 278	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))))
17	16	baibd 538	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
18	17	notbid 317	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
19	18	biimpd 228	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) → ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
20	19	expimpd 452	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) → ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
21		eldif 3956	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)))
22		fvex 6906	. . . . . . . 8 ⊢ (𝐹‘𝑥) ∈ V
23		eldifsn 4785	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (V ∖ {𝑍}) ↔ ((𝐹‘𝑥) ∈ V ∧ (𝐹‘𝑥) ≠ 𝑍))
24	22, 23	mpbiran 707	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (V ∖ {𝑍}) ↔ (𝐹‘𝑥) ≠ 𝑍)
25	24	necon2bbii 2982	. . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑍 ↔ ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))
26	20, 21, 25	3imtr4g 295	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → (𝐹‘𝑥) = 𝑍))
27	26	imp 405	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍)
28		suppss3.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑍) → 𝐵 = 𝑍)
29	3, 5, 27, 28	syl3anc 1368	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝐵 = 𝑍)
30	29, 7	suppss2 8207	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ (𝐹 supp 𝑍))
31	2, 30	eqsstrid 4027	1 ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  Vcvv 3462   ∖ cdif 3943   ⊆ wss 3946  {csn 4623   ↦ cmpt 5228  ^◡ccnv 5673   “ cima 5677   Fn wfn 6541  ‘cfv 6546  (class class class)co 7416   supp csupp 8166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-supp 8167

This theorem is referenced by:  evls1fldgencl  33562  eulerpartlems  34207