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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 4180 | . . 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 6435 | . . 3 ⊢ (𝜑 → {〈𝐴, 𝐶〉} = (𝑥 ∈ {𝐴} ↦ 𝐸)) |
7 | 6 | uneq1d 3766 | . 2 ⊢ (𝜑 → ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) = ((𝑥 ∈ {𝐴} ↦ 𝐸) ∪ {〈𝐵, 𝐷〉})) |
8 | fmptpr.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
9 | elex 3212 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
11 | fmptpr.4 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑌) | |
12 | elex 3212 | . . . 4 ⊢ (𝐷 ∈ 𝑌 → 𝐷 ∈ V) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
14 | df-pr 4180 | . . . . 5 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
15 | 14 | eqcomi 2631 | . . . 4 ⊢ ({𝐴} ∪ {𝐵}) = {𝐴, 𝐵} |
16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → ({𝐴} ∪ {𝐵}) = {𝐴, 𝐵}) |
17 | fmptpr.6 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐸 = 𝐷) | |
18 | 10, 13, 16, 17 | fmptapd 6437 | . 2 ⊢ (𝜑 → ((𝑥 ∈ {𝐴} ↦ 𝐸) ∪ {〈𝐵, 𝐷〉}) = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸)) |
19 | 2, 7, 18 | 3eqtrd 2660 | 1 ⊢ (𝜑 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∪ cun 3572 {csn 4177 {cpr 4179 〈cop 4183 ↦ cmpt 4729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-mpt 4730 |
This theorem is referenced by: pmtrprfvalrn 17908 esumsnf 30126 sge0sn 40596 zlmodzxzscm 42135 zlmodzxzadd 42136 |
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