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Mirrors > Home > MPE Home > Th. List > fmptap | Structured version Visualization version GIF version |
Description: Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fmptap.0a | ⊢ 𝐴 ∈ V |
fmptap.0b | ⊢ 𝐵 ∈ V |
fmptap.1 | ⊢ (𝑅 ∪ {𝐴}) = 𝑆 |
fmptap.2 | ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐵) |
Ref | Expression |
---|---|
fmptap | ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmptap.0a | . . . . 5 ⊢ 𝐴 ∈ V | |
2 | fmptap.0b | . . . . 5 ⊢ 𝐵 ∈ V | |
3 | fmptsn 7114 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵)) | |
4 | 1, 2, 3 | mp2an 691 | . . . 4 ⊢ {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵) |
5 | elsni 4604 | . . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
6 | fmptap.2 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐵) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝐶 = 𝐵) |
8 | 7 | mpteq2ia 5209 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ 𝐶) = (𝑥 ∈ {𝐴} ↦ 𝐵) |
9 | 4, 8 | eqtr4i 2764 | . . 3 ⊢ {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶) |
10 | 9 | uneq2i 4121 | . 2 ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)) |
11 | mptun 6648 | . 2 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)) | |
12 | fmptap.1 | . . 3 ⊢ (𝑅 ∪ {𝐴}) = 𝑆 | |
13 | 12 | mpteq1i 5202 | . 2 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶) |
14 | 10, 11, 13 | 3eqtr2i 2767 | 1 ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∪ cun 3909 {csn 4587 ⟨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-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 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-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 |
This theorem is referenced by: (None) |
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