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Mirrors > Home > ILE Home > Th. List > fmptapd | GIF version |
Description: Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) |
Ref | Expression |
---|---|
fmptapd.0a | ⊢ (𝜑 → 𝐴 ∈ V) |
fmptapd.0b | ⊢ (𝜑 → 𝐵 ∈ V) |
fmptapd.1 | ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆) |
fmptapd.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵) |
Ref | Expression |
---|---|
fmptapd | ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {〈𝐴, 𝐵〉}) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmptapd.0a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) | |
2 | fmptapd.0b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) | |
3 | fmptsn 5352 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {〈𝐴, 𝐵〉} = (𝑥 ∈ {𝐴} ↦ 𝐵)) | |
4 | 1, 2, 3 | syl2anc 391 | . . . 4 ⊢ (𝜑 → {〈𝐴, 𝐵〉} = (𝑥 ∈ {𝐴} ↦ 𝐵)) |
5 | elsni 3393 | . . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
6 | fmptapd.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵) | |
7 | 5, 6 | sylan2 270 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → 𝐶 = 𝐵) |
8 | 7 | mpteq2dva 3847 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ {𝐴} ↦ 𝐶) = (𝑥 ∈ {𝐴} ↦ 𝐵)) |
9 | 4, 8 | eqtr4d 2075 | . . 3 ⊢ (𝜑 → {〈𝐴, 𝐵〉} = (𝑥 ∈ {𝐴} ↦ 𝐶)) |
10 | 9 | uneq2d 3097 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {〈𝐴, 𝐵〉}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))) |
11 | mptun 5029 | . . 3 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)) | |
12 | 11 | a1i 9 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))) |
13 | fmptapd.1 | . . 3 ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆) | |
14 | 13 | mpteq1d 3842 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
15 | 10, 12, 14 | 3eqtr2d 2078 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {〈𝐴, 𝐵〉}) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ∪ cun 2915 {csn 3375 〈cop 3378 ↦ cmpt 3818 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 |
This theorem is referenced by: fmptpr 5355 |
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