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Mirrors > Home > ILE Home > Th. List > fvmpt | GIF version |
Description: Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.) |
Ref | Expression |
---|---|
fvmptg.1 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
fvmptg.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) |
fvmpt.3 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
fvmpt | ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmpt.3 | . 2 ⊢ 𝐶 ∈ V | |
2 | fvmptg.1 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
3 | fvmptg.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
4 | 2, 3 | fvmptg 5497 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → (𝐹‘𝐴) = 𝐶) |
5 | 1, 4 | mpan2 421 | 1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ↦ cmpt 3989 ‘cfv 5123 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 |
This theorem is referenced by: reldm 6084 rdg0 6284 oacl 6356 fvmptmap 6579 xpcomco 6720 uzval 9328 sqrtrval 10772 fsumcnv 11206 ege2le3 11377 qnumval 11863 qdenval 11864 peano4nninf 13200 peano3nninf 13201 nninfsellemeq 13210 |
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