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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 4590 | . . 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 7116 | . . 3 ⊢ (𝜑 → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐸)) |
7 | 6 | uneq1d 4123 | . 2 ⊢ (𝜑 → ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = ((𝑥 ∈ {𝐴} ↦ 𝐸) ∪ {⟨𝐵, 𝐷⟩})) |
8 | fmptpr.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
9 | fmptpr.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑌) | |
10 | df-pr 4590 | . . . . 5 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
11 | 10 | eqcomi 2742 | . . . 4 ⊢ ({𝐴} ∪ {𝐵}) = {𝐴, 𝐵} |
12 | 11 | a1i 11 | . . 3 ⊢ (𝜑 → ({𝐴} ∪ {𝐵}) = {𝐴, 𝐵}) |
13 | fmptpr.6 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐸 = 𝐷) | |
14 | 8, 9, 12, 13 | fmptapd 7118 | . 2 ⊢ (𝜑 → ((𝑥 ∈ {𝐴} ↦ 𝐸) ∪ {⟨𝐵, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸)) |
15 | 2, 7, 14 | 3eqtrd 2777 | 1 ⊢ (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∪ cun 3909 {csn 4587 {cpr 4589 ⟨cop 4593 ↦ cmpt 5189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-opab 5169 df-mpt 5190 |
This theorem is referenced by: pmtrprfvalrn 19275 esumsnf 32720 sge0sn 44706 zlmodzxzscm 46519 zlmodzxzadd 46520 |
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