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Mirrors > Home > ILE Home > Th. List > fvmptf | GIF version |
Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5612 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
fvmptf.1 | ⊢ Ⅎ𝑥𝐴 |
fvmptf.2 | ⊢ Ⅎ𝑥𝐶 |
fvmptf.3 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
fvmptf.4 | ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) |
Ref | Expression |
---|---|
fvmptf | ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2763 | . . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) | |
2 | fvmptf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | fvmptf.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 | |
4 | 3 | nfel1 2343 | . . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ V |
5 | fvmptf.4 | . . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
6 | nfmpt1 4111 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵) | |
7 | 5, 6 | nfcxfr 2329 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 |
8 | 7, 2 | nffv 5544 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝐴) |
9 | 8, 3 | nfeq 2340 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝐴) = 𝐶 |
10 | 4, 9 | nfim 1583 | . . . 4 ⊢ Ⅎ𝑥(𝐶 ∈ V → (𝐹‘𝐴) = 𝐶) |
11 | fvmptf.3 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
12 | 11 | eleq1d 2258 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V)) |
13 | fveq2 5534 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
14 | 13, 11 | eqeq12d 2204 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = 𝐵 ↔ (𝐹‘𝐴) = 𝐶)) |
15 | 12, 14 | imbi12d 234 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹‘𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶))) |
16 | 5 | fvmpt2 5619 | . . . . 5 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) = 𝐵) |
17 | 16 | ex 115 | . . . 4 ⊢ (𝑥 ∈ 𝐷 → (𝐵 ∈ V → (𝐹‘𝑥) = 𝐵)) |
18 | 2, 10, 15, 17 | vtoclgaf 2817 | . . 3 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)) |
19 | 1, 18 | syl5 32 | . 2 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ 𝑉 → (𝐹‘𝐴) = 𝐶)) |
20 | 19 | imp 124 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 Ⅎwnfc 2319 Vcvv 2752 ↦ cmpt 4079 ‘cfv 5235 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 |
This theorem is referenced by: fvmptd3 5629 elfvmptrab1 5630 sumrbdclem 11416 fsum3 11426 isumss 11430 prodrbdclem 11610 prodmodclem2a 11615 zproddc 11618 fprodntrivap 11623 prodssdc 11628 pcmpt 12374 |
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