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Mirrors > Home > MPE Home > Th. List > mpteq2iaOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mpteq2ia 5244 as of 11-Nov-2024. (Contributed by Mario Carneiro, 16-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mpteq2ia.1 | ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
mpteq2iaOLD | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | ax-gen 1789 | . 2 ⊢ ∀𝑥 𝐴 = 𝐴 |
3 | mpteq2ia.1 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) | |
4 | 3 | rgen 3057 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶 |
5 | mpteq12f 5229 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
6 | 2, 4, 5 | mp2an 689 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ↦ cmpt 5224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-opab 5204 df-mpt 5225 |
This theorem is referenced by: (None) |
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