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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 7176 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵)) | |
4 | 1, 2, 3 | mp2an 691 | . . . 4 ⊢ {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵) |
5 | elsni 4646 | . . . . . 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 5251 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ 𝐶) = (𝑥 ∈ {𝐴} ↦ 𝐵) |
9 | 4, 8 | eqtr4i 2759 | . . 3 ⊢ {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶) |
10 | 9 | uneq2i 4159 | . 2 ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)) |
11 | mptun 6701 | . 2 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)) | |
12 | fmptap.1 | . . 3 ⊢ (𝑅 ∪ {𝐴}) = 𝑆 | |
13 | 12 | mpteq1i 5244 | . 2 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶) |
14 | 10, 11, 13 | 3eqtr2i 2762 | 1 ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∪ cun 3945 {csn 4629 ⟨cop 4635 ↦ cmpt 5231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 |
This theorem is referenced by: (None) |
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