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Theorem suppdm 42828
Description: If the range of a function does not contain the zero, the support of the function equals its domain. (Contributed by AV, 20-May-2020.)
Assertion
Ref Expression
suppdm (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = dom 𝐹)

Proof of Theorem suppdm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 suppval1 7470 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍})
21adantr 472 . 2 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍})
3 df-nel 3036 . . . . . 6 (𝑍 ∉ ran 𝐹 ↔ ¬ 𝑍 ∈ ran 𝐹)
4 fvelrn 6516 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
543ad2antl1 1201 . . . . . . . 8 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
6 eleq1 2827 . . . . . . . . 9 (𝑍 = (𝐹𝑥) → (𝑍 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
76eqcoms 2768 . . . . . . . 8 ((𝐹𝑥) = 𝑍 → (𝑍 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
85, 7syl5ibrcom 237 . . . . . . 7 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = 𝑍𝑍 ∈ ran 𝐹))
98necon3bd 2946 . . . . . 6 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑍 ∈ ran 𝐹 → (𝐹𝑥) ≠ 𝑍))
103, 9syl5bi 232 . . . . 5 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝑍 ∉ ran 𝐹 → (𝐹𝑥) ≠ 𝑍))
1110impancom 455 . . . 4 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝑥 ∈ dom 𝐹 → (𝐹𝑥) ≠ 𝑍))
1211ralrimiv 3103 . . 3 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → ∀𝑥 ∈ dom 𝐹(𝐹𝑥) ≠ 𝑍)
13 rabid2 3257 . . 3 (dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍} ↔ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) ≠ 𝑍)
1412, 13sylibr 224 . 2 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍})
152, 14eqtr4d 2797 1 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wnel 3035  wral 3050  {crab 3054  dom cdm 5266  ran crn 5267  Fun wfun 6043  cfv 6049  (class class class)co 6814   supp csupp 7464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-supp 7465
This theorem is referenced by:  elbigolo1  42879
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