Theorem suppss3 32647

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 7397	. 2 ⊢ (𝐺 supp 𝑍) = ((𝑥 ∈ 𝐴 ↦ 𝐵) supp 𝑍)
3		simpl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝜑)
4		eldifi 4094	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴)
5	4	adantl 481	. . . 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 7191	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
9	6, 7, 8	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
10		suppss3.z	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
11		suppimacnv 8153	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
12	9, 10, 11	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
13	12	eleq2d 2814	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ 𝑥 ∈ (◡𝐹 “ (V ∖ {𝑍}))))
14		elpreima 7030	. . . . . . . . . . . 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 279	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))))
17	16	baibd 539	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
18	17	notbid 318	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
19	18	biimpd 229	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) → ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
20	19	expimpd 453	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) → ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
21		eldif 3924	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)))
22		fvex 6871	. . . . . . . 8 ⊢ (𝐹‘𝑥) ∈ V
23		eldifsn 4750	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (V ∖ {𝑍}) ↔ ((𝐹‘𝑥) ∈ V ∧ (𝐹‘𝑥) ≠ 𝑍))
24	22, 23	mpbiran 709	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (V ∖ {𝑍}) ↔ (𝐹‘𝑥) ≠ 𝑍)
25	24	necon2bbii 2976	. . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑍 ↔ ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))
26	20, 21, 25	3imtr4g 296	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → (𝐹‘𝑥) = 𝑍))
27	26	imp 406	. . . 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 1373	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝐵 = 𝑍)
30	29, 7	suppss2 8179	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ (𝐹 supp 𝑍))
31	2, 30	eqsstrid 3985	1 ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3447   ∖ cdif 3911   ⊆ wss 3914  {csn 4589   ↦ cmpt 5188  ^◡ccnv 5637   “ cima 5641   Fn wfn 6506  ‘cfv 6511  (class class class)co 7387   supp csupp 8139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-supp 8140

This theorem is referenced by:  evls1fldgencl  33665  eulerpartlems  34351