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 6713 | . . . . 5 ⊢ (𝑥 = {〈0, 𝐵〉} → (𝑥‘0) = ({〈0, 𝐵〉}‘0)) | |
4 | 3 | adantl 485 | . . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 = {〈0, 𝐵〉}) → (𝑥‘0) = ({〈0, 𝐵〉}‘0)) |
5 | c0ex 10824 | . . . . . . . 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 484 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 = {〈0, 𝐵〉}) → (0 ∈ V ∧ 𝐵 ∈ 𝑋)) |
9 | fvsng 6992 | . . . . 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 2777 | . . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 = {〈0, 𝐵〉}) → (𝑥‘0) = 𝐵) |
12 | 11 | fveq2d 6718 | . 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 488 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋) | |
15 | 13, 14 | fsnd 6700 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → {〈0, 𝐵〉}:{0}⟶𝑋) |
16 | snex 5321 | . . . . 5 ⊢ {0} ∈ V | |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝐵 ∈ 𝑋 → {0} ∈ V) |
18 | elmapg 8518 | . . . 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 6729 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → (𝐴‘𝐵) ∈ V) | |
22 | 2, 12, 20, 21 | fvmptd 6822 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → (𝐹‘{〈0, 𝐵〉}) = (𝐴‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3405 {csn 4538 〈cop 4544 ↦ cmpt 5132 ⟶wf 6373 ‘cfv 6377 (class class class)co 7210 ↑m cmap 8505 0cc0 10726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-mulcl 10788 ax-i2m1 10794 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3063 df-rex 3064 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-ov 7213 df-oprab 7214 df-mpo 7215 df-map 8507 |
This theorem is referenced by: (None) |
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