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Mirrors > Home > ILE Home > Th. List > fconst3m | GIF version |
Description: Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.) |
Ref | Expression |
---|---|
fconst3m | ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconstfvm 5697 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵))) | |
2 | fnfun 5279 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
3 | fndm 5281 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
4 | eqimss2 3192 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → 𝐴 ⊆ dom 𝐹) | |
5 | 3, 4 | syl 14 | . . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐴 ⊆ dom 𝐹) |
6 | funconstss 5597 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ {𝐵}))) | |
7 | 2, 5, 6 | syl2anc 409 | . . 3 ⊢ (𝐹 Fn 𝐴 → (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ {𝐵}))) |
8 | 7 | pm5.32i 450 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵}))) |
9 | 1, 8 | bitrdi 195 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∃wex 1479 ∈ wcel 2135 ∀wral 2442 ⊆ wss 3111 {csn 3570 ◡ccnv 4597 dom cdm 4598 “ cima 4601 Fun wfun 5176 Fn wfn 5177 ⟶wf 5178 ‘cfv 5182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fo 5188 df-fv 5190 |
This theorem is referenced by: fconst4m 5699 |
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