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Mirrors > Home > MPE Home > Th. List > suppval1 | Structured version Visualization version GIF version |
Description: The value of the operation constructing the support of a function. (Contributed by AV, 6-Apr-2019.) |
Ref | Expression |
---|---|
suppval1 | ⊢ ((Fun 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋‘𝑖) ≠ 𝑍}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppval 7532 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}}) | |
2 | 1 | 3adant1 1161 | . 2 ⊢ ((Fun 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}}) |
3 | funfn 6129 | . . . . . . . . 9 ⊢ (Fun 𝑋 ↔ 𝑋 Fn dom 𝑋) | |
4 | 3 | biimpi 208 | . . . . . . . 8 ⊢ (Fun 𝑋 → 𝑋 Fn dom 𝑋) |
5 | 4 | 3ad2ant1 1164 | . . . . . . 7 ⊢ ((Fun 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑋 Fn dom 𝑋) |
6 | fnsnfv 6481 | . . . . . . 7 ⊢ ((𝑋 Fn dom 𝑋 ∧ 𝑖 ∈ dom 𝑋) → {(𝑋‘𝑖)} = (𝑋 “ {𝑖})) | |
7 | 5, 6 | sylan 576 | . . . . . 6 ⊢ (((Fun 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑖 ∈ dom 𝑋) → {(𝑋‘𝑖)} = (𝑋 “ {𝑖})) |
8 | 7 | eqcomd 2803 | . . . . 5 ⊢ (((Fun 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑖 ∈ dom 𝑋) → (𝑋 “ {𝑖}) = {(𝑋‘𝑖)}) |
9 | 8 | neeq1d 3028 | . . . 4 ⊢ (((Fun 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑖 ∈ dom 𝑋) → ((𝑋 “ {𝑖}) ≠ {𝑍} ↔ {(𝑋‘𝑖)} ≠ {𝑍})) |
10 | fvex 6422 | . . . . . 6 ⊢ (𝑋‘𝑖) ∈ V | |
11 | sneqbg 4558 | . . . . . 6 ⊢ ((𝑋‘𝑖) ∈ V → ({(𝑋‘𝑖)} = {𝑍} ↔ (𝑋‘𝑖) = 𝑍)) | |
12 | 10, 11 | mp1i 13 | . . . . 5 ⊢ (((Fun 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑖 ∈ dom 𝑋) → ({(𝑋‘𝑖)} = {𝑍} ↔ (𝑋‘𝑖) = 𝑍)) |
13 | 12 | necon3bid 3013 | . . . 4 ⊢ (((Fun 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑖 ∈ dom 𝑋) → ({(𝑋‘𝑖)} ≠ {𝑍} ↔ (𝑋‘𝑖) ≠ 𝑍)) |
14 | 9, 13 | bitrd 271 | . . 3 ⊢ (((Fun 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑖 ∈ dom 𝑋) → ((𝑋 “ {𝑖}) ≠ {𝑍} ↔ (𝑋‘𝑖) ≠ 𝑍)) |
15 | 14 | rabbidva 3370 | . 2 ⊢ ((Fun 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋‘𝑖) ≠ 𝑍}) |
16 | 2, 15 | eqtrd 2831 | 1 ⊢ ((Fun 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋‘𝑖) ≠ 𝑍}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ≠ wne 2969 {crab 3091 Vcvv 3383 {csn 4366 dom cdm 5310 “ cima 5313 Fun wfun 6093 Fn wfn 6094 ‘cfv 6099 (class class class)co 6876 supp csupp 7530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-fv 6107 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-supp 7531 |
This theorem is referenced by: suppvalfn 7537 suppfnss 7555 suppfnssOLD 7556 fnsuppres 7558 domnmsuppn0 42937 rmsuppss 42938 mndpsuppss 42939 scmsuppss 42940 suppdm 43087 |
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