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Mirrors > Home > ILE Home > Th. List > fconst4m | GIF version |
Description: Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.) |
Ref | Expression |
---|---|
fconst4m | ⊢ (∃x x ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ (◡𝐹 “ {B}) = A))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconst3m 5323 | . 2 ⊢ (∃x x ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ A ⊆ (◡𝐹 “ {B})))) | |
2 | cnvimass 4631 | . . . . . 6 ⊢ (◡𝐹 “ {B}) ⊆ dom 𝐹 | |
3 | fndm 4941 | . . . . . 6 ⊢ (𝐹 Fn A → dom 𝐹 = A) | |
4 | 2, 3 | syl5sseq 2987 | . . . . 5 ⊢ (𝐹 Fn A → (◡𝐹 “ {B}) ⊆ A) |
5 | 4 | biantrurd 289 | . . . 4 ⊢ (𝐹 Fn A → (A ⊆ (◡𝐹 “ {B}) ↔ ((◡𝐹 “ {B}) ⊆ A ∧ A ⊆ (◡𝐹 “ {B})))) |
6 | eqss 2954 | . . . 4 ⊢ ((◡𝐹 “ {B}) = A ↔ ((◡𝐹 “ {B}) ⊆ A ∧ A ⊆ (◡𝐹 “ {B}))) | |
7 | 5, 6 | syl6bbr 187 | . . 3 ⊢ (𝐹 Fn A → (A ⊆ (◡𝐹 “ {B}) ↔ (◡𝐹 “ {B}) = A)) |
8 | 7 | pm5.32i 427 | . 2 ⊢ ((𝐹 Fn A ∧ A ⊆ (◡𝐹 “ {B})) ↔ (𝐹 Fn A ∧ (◡𝐹 “ {B}) = A)) |
9 | 1, 8 | syl6bb 185 | 1 ⊢ (∃x x ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ (◡𝐹 “ {B}) = A))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ⊆ wss 2911 {csn 3367 ◡ccnv 4287 dom cdm 4288 “ cima 4291 Fn wfn 4840 ⟶wf 4841 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fo 4851 df-fv 4853 |
This theorem is referenced by: (None) |
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