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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-mpt2mptALT | Structured version Visualization version GIF version |
Description: Alternate proof of mpt2mpt 6794. (Contributed by BJ, 30-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-mpt2mptALT.1 | ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
bj-mpt2mptALT | ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp2 5166 | . . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉) | |
2 | 1 | anbi1i 731 | . . . 4 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶)) |
3 | r19.41v 3118 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶)) | |
4 | r19.41v 3118 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ (∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶)) | |
5 | bj-mpt2mptALT.1 | . . . . . . . . 9 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝐶 = 𝐷) | |
6 | 5 | eqeq2d 2661 | . . . . . . . 8 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑡 = 𝐶 ↔ 𝑡 = 𝐷)) |
7 | 6 | pm5.32i 670 | . . . . . . 7 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)) |
8 | 7 | rexbii 3070 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)) |
9 | 4, 8 | bitr3i 266 | . . . . 5 ⊢ ((∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)) |
10 | 9 | rexbii 3070 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)) |
11 | 2, 3, 10 | 3bitr2i 288 | . . 3 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)) |
12 | 11 | opabbii 4750 | . 2 ⊢ {〈𝑧, 𝑡〉 ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)} = {〈𝑧, 𝑡〉 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)} |
13 | df-mpt 4763 | . 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = {〈𝑧, 𝑡〉 ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)} | |
14 | bj-dfmpt2a 33196 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = {〈𝑧, 𝑡〉 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)} | |
15 | 12, 13, 14 | 3eqtr4i 2683 | 1 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∃wrex 2942 〈cop 4216 {copab 4745 ↦ cmpt 4762 × cxp 5141 ↦ cmpt2 6692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-opab 4746 df-mpt 4763 df-xp 5149 df-oprab 6694 df-mpt2 6695 |
This theorem is referenced by: (None) |
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