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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 5505 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → (𝐹‘𝐴) = 𝐶) |
5 | 1, 4 | mpan2 422 | 1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ↦ cmpt 3997 ‘cfv 5131 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 |
This theorem is referenced by: reldm 6092 rdg0 6292 oacl 6364 fvmptmap 6587 xpcomco 6728 uzval 9352 sqrtrval 10804 fsumcnv 11238 ege2le3 11414 qnumval 11899 qdenval 11900 peano4nninf 13375 peano3nninf 13376 nninfsellemeq 13385 |
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