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Mirrors > Home > MPE Home > Th. List > fmptpr | Structured version Visualization version GIF version |
Description: Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.) |
Ref | Expression |
---|---|
fmptpr.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
fmptpr.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
fmptpr.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
fmptpr.4 | ⊢ (𝜑 → 𝐷 ∈ 𝑌) |
fmptpr.5 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐸 = 𝐶) |
fmptpr.6 | ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐸 = 𝐷) |
Ref | Expression |
---|---|
fmptpr | ⊢ (𝜑 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4569 | . . 3 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})) |
3 | fmptpr.5 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐸 = 𝐶) | |
4 | fmptpr.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | fmptpr.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
6 | 3, 4, 5 | fmptsnd 6930 | . . 3 ⊢ (𝜑 → {〈𝐴, 𝐶〉} = (𝑥 ∈ {𝐴} ↦ 𝐸)) |
7 | 6 | uneq1d 4137 | . 2 ⊢ (𝜑 → ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) = ((𝑥 ∈ {𝐴} ↦ 𝐸) ∪ {〈𝐵, 𝐷〉})) |
8 | fmptpr.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
9 | 8 | elexd 3514 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
10 | fmptpr.4 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑌) | |
11 | 10 | elexd 3514 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
12 | df-pr 4569 | . . . . 5 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
13 | 12 | eqcomi 2830 | . . . 4 ⊢ ({𝐴} ∪ {𝐵}) = {𝐴, 𝐵} |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → ({𝐴} ∪ {𝐵}) = {𝐴, 𝐵}) |
15 | fmptpr.6 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐸 = 𝐷) | |
16 | 9, 11, 14, 15 | fmptapd 6932 | . 2 ⊢ (𝜑 → ((𝑥 ∈ {𝐴} ↦ 𝐸) ∪ {〈𝐵, 𝐷〉}) = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸)) |
17 | 2, 7, 16 | 3eqtrd 2860 | 1 ⊢ (𝜑 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∪ cun 3933 {csn 4566 {cpr 4568 〈cop 4572 ↦ cmpt 5145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-opab 5128 df-mpt 5146 |
This theorem is referenced by: pmtrprfvalrn 18615 esumsnf 31323 sge0sn 42660 zlmodzxzscm 44404 zlmodzxzadd 44405 |
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