Theorem elsuppfng 8180

Description: An element of the support of a function with a given domain. This version of elsuppfn 8181 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-rep 5289. (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 8178	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍})
2	1	eleq2d 2815	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ 𝑆 ∈ {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍}))
3		fveq2 6902	. . . 4 ⊢ (𝑖 = 𝑆 → (𝐹‘𝑖) = (𝐹‘𝑆))
4	3	neeq1d 2997	. . 3 ⊢ (𝑖 = 𝑆 → ((𝐹‘𝑖) ≠ 𝑍 ↔ (𝐹‘𝑆) ≠ 𝑍))
5	4	elrab 3684	. 2 ⊢ (𝑆 ∈ {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍} ↔ (𝑆 ∈ 𝑋 ∧ (𝐹‘𝑆) ≠ 𝑍))
6	2, 5	bitrdi 286	1 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆 ∈ 𝑋 ∧ (𝐹‘𝑆) ≠ 𝑍)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2937  {crab 3430   Fn wfn 6548  ‘cfv 6553  (class class class)co 7426   supp csupp 8171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-supp 8172

This theorem is referenced by:  suppss  8205  suppssrg  8208  ciclcl  17792  cicrcl  17793  ismhp3  22074  mdegleb  26020  suppiniseg  32487