Theorem elsuppfng 8168

Description: An element of the support of a function with a given domain. This version of elsuppfn 8169 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-rep 5249. (Contributed by SN, 5-Aug-2024.)

Assertion

Ref	Expression
elsuppfng	⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆 ∈ 𝑋 ∧ (𝐹‘𝑆) ≠ 𝑍)))

Proof of Theorem elsuppfng

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppvalfng 8166	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍})
2	1	eleq2d 2820	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ 𝑆 ∈ {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍}))
3		fveq2 6876	. . . 4 ⊢ (𝑖 = 𝑆 → (𝐹‘𝑖) = (𝐹‘𝑆))
4	3	neeq1d 2991	. . 3 ⊢ (𝑖 = 𝑆 → ((𝐹‘𝑖) ≠ 𝑍 ↔ (𝐹‘𝑆) ≠ 𝑍))
5	4	elrab 3671	. 2 ⊢ (𝑆 ∈ {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍} ↔ (𝑆 ∈ 𝑋 ∧ (𝐹‘𝑆) ≠ 𝑍))
6	2, 5	bitrdi 287	1 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆 ∈ 𝑋 ∧ (𝐹‘𝑆) ≠ 𝑍)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  {crab 3415   Fn wfn 6526  ‘cfv 6531  (class class class)co 7405   supp csupp 8159

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-supp 8160

This theorem is referenced by:  suppss  8193  suppssrg  8195  ciclcl  17815  cicrcl  17816  ismhp3  22080  mdegleb  26021  suppiniseg  32663