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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 5572 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 2741 | . . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) | |
2 | fvmptf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | fvmptf.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 | |
4 | 3 | nfel1 2323 | . . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ V |
5 | fvmptf.4 | . . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
6 | nfmpt1 4082 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵) | |
7 | 5, 6 | nfcxfr 2309 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 |
8 | 7, 2 | nffv 5506 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝐴) |
9 | 8, 3 | nfeq 2320 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝐴) = 𝐶 |
10 | 4, 9 | nfim 1565 | . . . 4 ⊢ Ⅎ𝑥(𝐶 ∈ V → (𝐹‘𝐴) = 𝐶) |
11 | fvmptf.3 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
12 | 11 | eleq1d 2239 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V)) |
13 | fveq2 5496 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
14 | 13, 11 | eqeq12d 2185 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = 𝐵 ↔ (𝐹‘𝐴) = 𝐶)) |
15 | 12, 14 | imbi12d 233 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹‘𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶))) |
16 | 5 | fvmpt2 5579 | . . . . 5 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) = 𝐵) |
17 | 16 | ex 114 | . . . 4 ⊢ (𝑥 ∈ 𝐷 → (𝐵 ∈ V → (𝐹‘𝑥) = 𝐵)) |
18 | 2, 10, 15, 17 | vtoclgaf 2795 | . . 3 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)) |
19 | 1, 18 | syl5 32 | . 2 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ 𝑉 → (𝐹‘𝐴) = 𝐶)) |
20 | 19 | imp 123 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 Ⅎwnfc 2299 Vcvv 2730 ↦ cmpt 4050 ‘cfv 5198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 |
This theorem is referenced by: fvmptd3 5589 elfvmptrab1 5590 sumrbdclem 11340 fsum3 11350 isumss 11354 prodrbdclem 11534 prodmodclem2a 11539 zproddc 11542 fprodntrivap 11547 prodssdc 11552 pcmpt 12295 |
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