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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-mpomptALT | Structured version Visualization version GIF version |
Description: Alternate proof of mpompt 7259. (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 5572 | . . . . 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 3346 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐶)) | |
4 | r19.41v 3346 | . . . . . 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 2831 | . . . . . . . 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 3246 | . . . . . 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 3246 | . . . 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 5126 | . 2 ⊢ {〈𝑧, 𝑡〉 ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)} = {〈𝑧, 𝑡〉 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)} |
13 | df-mpt 5140 | . 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = {〈𝑧, 𝑡〉 ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)} | |
14 | bj-dfmpoa 34434 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = {〈𝑧, 𝑡〉 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝑡 = 𝐷)} | |
15 | 12, 13, 14 | 3eqtr4i 2853 | 1 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 〈cop 4566 {copab 5121 ↦ cmpt 5139 × cxp 5546 ∈ cmpo 7151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5122 df-mpt 5140 df-xp 5554 df-oprab 7153 df-mpo 7154 |
This theorem is referenced by: (None) |
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