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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 5191 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 2560 | . . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) | |
2 | fvmptf.1 | . . . 4 ⊢ ℲxA | |
3 | fvmptf.2 | . . . . . 6 ⊢ Ⅎx𝐶 | |
4 | 3 | nfel1 2185 | . . . . 5 ⊢ Ⅎx 𝐶 ∈ V |
5 | fvmptf.4 | . . . . . . . 8 ⊢ 𝐹 = (x ∈ 𝐷 ↦ B) | |
6 | nfmpt1 3841 | . . . . . . . 8 ⊢ Ⅎx(x ∈ 𝐷 ↦ B) | |
7 | 5, 6 | nfcxfr 2172 | . . . . . . 7 ⊢ Ⅎx𝐹 |
8 | 7, 2 | nffv 5128 | . . . . . 6 ⊢ Ⅎx(𝐹‘A) |
9 | 8, 3 | nfeq 2182 | . . . . 5 ⊢ Ⅎx(𝐹‘A) = 𝐶 |
10 | 4, 9 | nfim 1461 | . . . 4 ⊢ Ⅎx(𝐶 ∈ V → (𝐹‘A) = 𝐶) |
11 | fvmptf.3 | . . . . . 6 ⊢ (x = A → B = 𝐶) | |
12 | 11 | eleq1d 2103 | . . . . 5 ⊢ (x = A → (B ∈ V ↔ 𝐶 ∈ V)) |
13 | fveq2 5121 | . . . . . 6 ⊢ (x = A → (𝐹‘x) = (𝐹‘A)) | |
14 | 13, 11 | eqeq12d 2051 | . . . . 5 ⊢ (x = A → ((𝐹‘x) = B ↔ (𝐹‘A) = 𝐶)) |
15 | 12, 14 | imbi12d 223 | . . . 4 ⊢ (x = A → ((B ∈ V → (𝐹‘x) = B) ↔ (𝐶 ∈ V → (𝐹‘A) = 𝐶))) |
16 | 5 | fvmpt2 5197 | . . . . 5 ⊢ ((x ∈ 𝐷 ∧ B ∈ V) → (𝐹‘x) = B) |
17 | 16 | ex 108 | . . . 4 ⊢ (x ∈ 𝐷 → (B ∈ V → (𝐹‘x) = B)) |
18 | 2, 10, 15, 17 | vtoclgaf 2612 | . . 3 ⊢ (A ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘A) = 𝐶)) |
19 | 1, 18 | syl5 28 | . 2 ⊢ (A ∈ 𝐷 → (𝐶 ∈ 𝑉 → (𝐹‘A) = 𝐶)) |
20 | 19 | imp 115 | 1 ⊢ ((A ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘A) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 Ⅎwnfc 2162 Vcvv 2551 ↦ cmpt 3809 ‘cfv 4845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 |
This theorem is referenced by: (None) |
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