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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-mpomptALT | Structured version Visualization version GIF version |
Description: Alternate proof of mpompt 7528. (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 5696 | . . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉) | |
2 | 1 | anbi1i 623 | . . . 4 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶)) |
3 | r19.41v 3184 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶)) | |
4 | r19.41v 3184 | . . . . . 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 2739 | . . . . . . . 8 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑡 = 𝐶 ↔ 𝑡 = 𝐷)) |
7 | 6 | pm5.32i 574 | . . . . . . 7 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)) |
8 | 7 | rexbii 3090 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)) |
9 | 4, 8 | bitr3i 277 | . . . . 5 ⊢ ((∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)) |
10 | 9 | rexbii 3090 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)) |
11 | 2, 3, 10 | 3bitr2i 299 | . . 3 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)) |
12 | 11 | opabbii 5209 | . 2 ⊢ {〈𝑧, 𝑡〉 ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)} = {〈𝑧, 𝑡〉 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)} |
13 | df-mpt 5226 | . 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = {〈𝑧, 𝑡〉 ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)} | |
14 | bj-dfmpoa 36591 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = {〈𝑧, 𝑡〉 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)} | |
15 | 12, 13, 14 | 3eqtr4i 2766 | 1 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3066 〈cop 4630 {copab 5204 ↦ cmpt 5225 × cxp 5670 ∈ cmpo 7416 |
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-11 2147 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
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-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-opab 5205 df-mpt 5226 df-xp 5678 df-oprab 7418 df-mpo 7419 |
This theorem is referenced by: (None) |
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