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Mirrors > Home > ILE Home > Th. List > funconstss | GIF version |
Description: Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.) |
Ref | Expression |
---|---|
funconstss | ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ {𝐵}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funimass4 5531 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ {𝐵} ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ {𝐵})) | |
2 | funimass3 5595 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ {𝐵} ↔ 𝐴 ⊆ (◡𝐹 “ {𝐵}))) | |
3 | ssel2 3132 | . . . . . 6 ⊢ ((𝐴 ⊆ dom 𝐹 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹) | |
4 | 3 | anim2i 340 | . . . . 5 ⊢ ((Fun 𝐹 ∧ (𝐴 ⊆ dom 𝐹 ∧ 𝑥 ∈ 𝐴)) → (Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹)) |
5 | 4 | anassrs 398 | . . . 4 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ 𝑥 ∈ 𝐴) → (Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹)) |
6 | funfvex 5497 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V) | |
7 | elsng 3585 | . . . 4 ⊢ ((𝐹‘𝑥) ∈ V → ((𝐹‘𝑥) ∈ {𝐵} ↔ (𝐹‘𝑥) = 𝐵)) | |
8 | 5, 6, 7 | 3syl 17 | . . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ∈ {𝐵} ↔ (𝐹‘𝑥) = 𝐵)) |
9 | 8 | ralbidva 2460 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ {𝐵} ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)) |
10 | 1, 2, 9 | 3bitr3rd 218 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ {𝐵}))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 ∀wral 2442 Vcvv 2721 ⊆ wss 3111 {csn 3570 ◡ccnv 4597 dom cdm 4598 “ cima 4601 Fun wfun 5176 ‘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-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-fv 5190 |
This theorem is referenced by: fconst3m 5698 |
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