Theorem suppcofnd 7871

Description: The support of the composition of two functions. (Contributed by SN, 15-Sep-2023.)

Hypotheses

Ref	Expression
suppcofnd.z	⊢ (𝜑 → 𝑍 ∈ 𝑈)
suppcofnd.f	⊢ (𝜑 → 𝐹 Fn 𝐴)
suppcofnd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
suppcofnd.g	⊢ (𝜑 → 𝐺 Fn 𝐵)
suppcofnd.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
suppcofnd	⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) = {𝑥 ∈ 𝐵 ∣ ((𝐺‘𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑥)) ≠ 𝑍)})

Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝑥,𝑍   𝑥,𝑈   𝑥,𝑊   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem suppcofnd

Step	Hyp	Ref	Expression
1		suppcofnd.f	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		suppcofnd.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3	1, 2	fnexd 6981	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
4		suppcofnd.g	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵)
5		suppcofnd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
6	4, 5	fnexd 6981	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
7		suppco 7870	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
8	3, 6, 7	syl2anc 586	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
9		fncnvima2 6831	. . 3 ⊢ (𝐺 Fn 𝐵 → (◡𝐺 “ (𝐹 supp 𝑍)) = {𝑥 ∈ 𝐵 ∣ (𝐺‘𝑥) ∈ (𝐹 supp 𝑍)})
10	4, 9	syl 17	. 2 ⊢ (𝜑 → (◡𝐺 “ (𝐹 supp 𝑍)) = {𝑥 ∈ 𝐵 ∣ (𝐺‘𝑥) ∈ (𝐹 supp 𝑍)})
11		suppcofnd.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
12		elsuppfn 7838	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → ((𝐺‘𝑥) ∈ (𝐹 supp 𝑍) ↔ ((𝐺‘𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑥)) ≠ 𝑍)))
13	1, 2, 11, 12	syl3anc 1367	. . 3 ⊢ (𝜑 → ((𝐺‘𝑥) ∈ (𝐹 supp 𝑍) ↔ ((𝐺‘𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑥)) ≠ 𝑍)))
14	13	rabbidv 3480	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ (𝐺‘𝑥) ∈ (𝐹 supp 𝑍)} = {𝑥 ∈ 𝐵 ∣ ((𝐺‘𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑥)) ≠ 𝑍)})
15	8, 10, 14	3eqtrd 2860	1 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) = {𝑥 ∈ 𝐵 ∣ ((𝐺‘𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑥)) ≠ 𝑍)})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  {crab 3142  Vcvv 3494  ^◡ccnv 5554   “ cima 5558   ∘ ccom 5559   Fn wfn 6350  ‘cfv 6355  (class class class)co 7156   supp csupp 7830

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-supp 7831

This theorem is referenced by:  mhpinvcl  20339