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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-mpomptALT | Structured version Visualization version GIF version |
Description: Alternate proof of mpompt 7266. (Contributed by BJ, 30-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-mpomptALT.1 | ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
bj-mpomptALT | ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp2 5579 | . . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉) | |
2 | 1 | anbi1i 625 | . . . 4 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶)) |
3 | r19.41v 3347 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶)) | |
4 | r19.41v 3347 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ (∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶)) | |
5 | bj-mpomptALT.1 | . . . . . . . . 9 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝐶 = 𝐷) | |
6 | 5 | eqeq2d 2832 | . . . . . . . 8 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑡 = 𝐶 ↔ 𝑡 = 𝐷)) |
7 | 6 | pm5.32i 577 | . . . . . . 7 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)) |
8 | 7 | rexbii 3247 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)) |
9 | 4, 8 | bitr3i 279 | . . . . 5 ⊢ ((∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)) |
10 | 9 | rexbii 3247 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)) |
11 | 2, 3, 10 | 3bitr2i 301 | . . 3 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)) |
12 | 11 | opabbii 5133 | . 2 ⊢ {〈𝑧, 𝑡〉 ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)} = {〈𝑧, 𝑡〉 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)} |
13 | df-mpt 5147 | . 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = {〈𝑧, 𝑡〉 ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)} | |
14 | bj-dfmpoa 34413 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = {〈𝑧, 𝑡〉 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)} | |
15 | 12, 13, 14 | 3eqtr4i 2854 | 1 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 〈cop 4573 {copab 5128 ↦ cmpt 5146 × cxp 5553 ∈ cmpo 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-mpt 5147 df-xp 5561 df-oprab 7160 df-mpo 7161 |
This theorem is referenced by: (None) |
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