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Mirrors > Home > MPE Home > Th. List > fnmptfvd | Structured version Visualization version GIF version |
Description: A function with a given domain is a mapping defined by its function values. (Contributed by AV, 1-Mar-2019.) |
Ref | Expression |
---|---|
fnmptfvd.m | ⊢ (𝜑 → 𝑀 Fn 𝐴) |
fnmptfvd.s | ⊢ (𝑖 = 𝑎 → 𝐷 = 𝐶) |
fnmptfvd.d | ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝐷 ∈ 𝑈) |
fnmptfvd.c | ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ 𝑉) |
Ref | Expression |
---|---|
fnmptfvd | ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmptfvd.m | . . 3 ⊢ (𝜑 → 𝑀 Fn 𝐴) | |
2 | fnmptfvd.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ 𝑉) | |
3 | 2 | ralrimiva 3149 | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 𝐶 ∈ 𝑉) |
4 | eqid 2798 | . . . . 5 ⊢ (𝑎 ∈ 𝐴 ↦ 𝐶) = (𝑎 ∈ 𝐴 ↦ 𝐶) | |
5 | 4 | fnmpt 6460 | . . . 4 ⊢ (∀𝑎 ∈ 𝐴 𝐶 ∈ 𝑉 → (𝑎 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) |
6 | 3, 5 | syl 17 | . . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) |
7 | eqfnfv 6779 | . . 3 ⊢ ((𝑀 Fn 𝐴 ∧ (𝑎 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖))) | |
8 | 1, 6, 7 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖))) |
9 | fnmptfvd.s | . . . . . . . 8 ⊢ (𝑖 = 𝑎 → 𝐷 = 𝐶) | |
10 | 9 | cbvmptv 5133 | . . . . . . 7 ⊢ (𝑖 ∈ 𝐴 ↦ 𝐷) = (𝑎 ∈ 𝐴 ↦ 𝐶) |
11 | 10 | eqcomi 2807 | . . . . . 6 ⊢ (𝑎 ∈ 𝐴 ↦ 𝐶) = (𝑖 ∈ 𝐴 ↦ 𝐷) |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ 𝐶) = (𝑖 ∈ 𝐴 ↦ 𝐷)) |
13 | 12 | fveq1d 6647 | . . . 4 ⊢ (𝜑 → ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖)) |
14 | 13 | eqeq2d 2809 | . . 3 ⊢ (𝜑 → ((𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖) ↔ (𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖))) |
15 | 14 | ralbidv 3162 | . 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖))) |
16 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝑖 ∈ 𝐴) | |
17 | fnmptfvd.d | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝐷 ∈ 𝑈) | |
18 | eqid 2798 | . . . . . 6 ⊢ (𝑖 ∈ 𝐴 ↦ 𝐷) = (𝑖 ∈ 𝐴 ↦ 𝐷) | |
19 | 18 | fvmpt2 6756 | . . . . 5 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝐷 ∈ 𝑈) → ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) = 𝐷) |
20 | 16, 17, 19 | syl2anc 587 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) = 𝐷) |
21 | 20 | eqeq2d 2809 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ((𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) ↔ (𝑀‘𝑖) = 𝐷)) |
22 | 21 | ralbidva 3161 | . 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷)) |
23 | 8, 15, 22 | 3bitrd 308 | 1 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ↦ cmpt 5110 Fn wfn 6319 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 |
This theorem is referenced by: cramerlem1 21292 dssmapnvod 40721 |
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