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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1arympt1fv | Structured version Visualization version GIF version |
Description: The value of a unary (endo)function in maps-to notation. (Contributed by AV, 16-May-2024.) |
Ref | Expression |
---|---|
1arympt1.f | ⊢ 𝐹 = (𝑥 ∈ (𝑋 ↑m {0}) ↦ (𝐴‘(𝑥‘0))) |
Ref | Expression |
---|---|
1arympt1fv | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → (𝐹‘{〈0, 𝐵〉}) = (𝐴‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1arympt1.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ (𝑋 ↑m {0}) ↦ (𝐴‘(𝑥‘0))) | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → 𝐹 = (𝑥 ∈ (𝑋 ↑m {0}) ↦ (𝐴‘(𝑥‘0)))) |
3 | fveq1 6650 | . . . . 5 ⊢ (𝑥 = {〈0, 𝐵〉} → (𝑥‘0) = ({〈0, 𝐵〉}‘0)) | |
4 | 3 | adantl 486 | . . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 = {〈0, 𝐵〉}) → (𝑥‘0) = ({〈0, 𝐵〉}‘0)) |
5 | c0ex 10658 | . . . . . . . 8 ⊢ 0 ∈ V | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 0 ∈ V) |
7 | 6 | anim1i 618 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → (0 ∈ V ∧ 𝐵 ∈ 𝑋)) |
8 | 7 | adantr 485 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 = {〈0, 𝐵〉}) → (0 ∈ V ∧ 𝐵 ∈ 𝑋)) |
9 | fvsng 6926 | . . . . 5 ⊢ ((0 ∈ V ∧ 𝐵 ∈ 𝑋) → ({〈0, 𝐵〉}‘0) = 𝐵) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 = {〈0, 𝐵〉}) → ({〈0, 𝐵〉}‘0) = 𝐵) |
11 | 4, 10 | eqtrd 2794 | . . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 = {〈0, 𝐵〉}) → (𝑥‘0) = 𝐵) |
12 | 11 | fveq2d 6655 | . 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 = {〈0, 𝐵〉}) → (𝐴‘(𝑥‘0)) = (𝐴‘𝐵)) |
13 | 5 | a1i 11 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → 0 ∈ V) |
14 | simpr 489 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋) | |
15 | 13, 14 | fsnd 6637 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → {〈0, 𝐵〉}:{0}⟶𝑋) |
16 | snex 5293 | . . . . 5 ⊢ {0} ∈ V | |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝐵 ∈ 𝑋 → {0} ∈ V) |
18 | elmapg 8422 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ {0} ∈ V) → ({〈0, 𝐵〉} ∈ (𝑋 ↑m {0}) ↔ {〈0, 𝐵〉}:{0}⟶𝑋)) | |
19 | 17, 18 | sylan2 596 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → ({〈0, 𝐵〉} ∈ (𝑋 ↑m {0}) ↔ {〈0, 𝐵〉}:{0}⟶𝑋)) |
20 | 15, 19 | mpbird 260 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → {〈0, 𝐵〉} ∈ (𝑋 ↑m {0})) |
21 | fvexd 6666 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → (𝐴‘𝐵) ∈ V) | |
22 | 2, 12, 20, 21 | fvmptd 6759 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → (𝐹‘{〈0, 𝐵〉}) = (𝐴‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1539 ∈ wcel 2112 Vcvv 3407 {csn 4515 〈cop 4521 ↦ cmpt 5105 ⟶wf 6324 ‘cfv 6328 (class class class)co 7143 ↑m cmap 8409 0cc0 10560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-mulcl 10622 ax-i2m1 10628 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-map 8411 |
This theorem is referenced by: (None) |
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