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Mirrors > Home > MPE Home > Th. List > mpteq12df | Structured version Visualization version GIF version |
Description: An equality inference for the maps-to notation. Compare mpteq12dv 5112. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
mpteq12df.1 | ⊢ Ⅎ𝑥𝜑 |
mpteq12df.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
mpteq12df.3 | ⊢ (𝜑 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
mpteq12df | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpteq12df.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1920 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | mpteq12df.2 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐶) | |
4 | 3 | eleq2d 2818 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶)) |
5 | mpteq12df.3 | . . . . 5 ⊢ (𝜑 → 𝐵 = 𝐷) | |
6 | 5 | eqeq2d 2749 | . . . 4 ⊢ (𝜑 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐷)) |
7 | 4, 6 | anbi12d 634 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷))) |
8 | 1, 2, 7 | opabbid 5092 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)}) |
9 | df-mpt 5108 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
10 | df-mpt 5108 | . 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐷) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)} | |
11 | 8, 9, 10 | 3eqtr4g 2798 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 Ⅎwnf 1790 ∈ wcel 2113 {copab 5089 ↦ cmpt 5107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-opab 5090 df-mpt 5108 |
This theorem is referenced by: mpteq12dv 5112 esumrnmpt2 31598 exrecfnlem 35162 smflimmpt 43866 |
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