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Mirrors > Home > ILE Home > Th. List > fnmptfvd | 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 2548 | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 𝐶 ∈ 𝑉) |
4 | eqid 2175 | . . . . 5 ⊢ (𝑎 ∈ 𝐴 ↦ 𝐶) = (𝑎 ∈ 𝐴 ↦ 𝐶) | |
5 | 4 | fnmpt 5334 | . . . 4 ⊢ (∀𝑎 ∈ 𝐴 𝐶 ∈ 𝑉 → (𝑎 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) |
6 | 3, 5 | syl 14 | . . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) |
7 | eqfnfv 5605 | . . 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 411 | . 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 4094 | . . . . . . 7 ⊢ (𝑖 ∈ 𝐴 ↦ 𝐷) = (𝑎 ∈ 𝐴 ↦ 𝐶) |
11 | 10 | eqcomi 2179 | . . . . . 6 ⊢ (𝑎 ∈ 𝐴 ↦ 𝐶) = (𝑖 ∈ 𝐴 ↦ 𝐷) |
12 | 11 | a1i 9 | . . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ 𝐶) = (𝑖 ∈ 𝐴 ↦ 𝐷)) |
13 | 12 | fveq1d 5509 | . . . 4 ⊢ (𝜑 → ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖)) |
14 | 13 | eqeq2d 2187 | . . 3 ⊢ (𝜑 → ((𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖) ↔ (𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖))) |
15 | 14 | ralbidv 2475 | . 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖))) |
16 | simpr 110 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝑖 ∈ 𝐴) | |
17 | fnmptfvd.d | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝐷 ∈ 𝑈) | |
18 | eqid 2175 | . . . . . 6 ⊢ (𝑖 ∈ 𝐴 ↦ 𝐷) = (𝑖 ∈ 𝐴 ↦ 𝐷) | |
19 | 18 | fvmpt2 5591 | . . . . 5 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝐷 ∈ 𝑈) → ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) = 𝐷) |
20 | 16, 17, 19 | syl2anc 411 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) = 𝐷) |
21 | 20 | eqeq2d 2187 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ((𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) ↔ (𝑀‘𝑖) = 𝐷)) |
22 | 21 | ralbidva 2471 | . 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷)) |
23 | 8, 15, 22 | 3bitrd 214 | 1 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2146 ∀wral 2453 ↦ cmpt 4059 Fn wfn 5203 ‘cfv 5208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-v 2737 df-sbc 2961 df-csb 3056 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-iota 5170 df-fun 5210 df-fn 5211 df-fv 5216 |
This theorem is referenced by: nninfdcinf 7159 nninfwlporlemd 7160 nninfwlporlem 7161 |
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