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| Mirrors > Home > MPE Home > Th. List > relopabiALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of relopabi 5832 (shorter but uses more axioms). (Contributed by Mario Carneiro, 21-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| relopabi.1 | ⊢ 𝐴 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
| Ref | Expression |
|---|---|
| relopabiALT | ⊢ Rel 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | relopabi.1 | . . . 4 ⊢ 𝐴 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 2 | df-opab 5206 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
| 3 | 1, 2 | eqtri 2765 | . . 3 ⊢ 𝐴 = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
| 4 | vex 3484 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
| 5 | vex 3484 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
| 6 | 4, 5 | opelvv 5725 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ (V × V) |
| 7 | eleq1 2829 | . . . . . . 7 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑧 ∈ (V × V) ↔ 〈𝑥, 𝑦〉 ∈ (V × V))) | |
| 8 | 6, 7 | mpbiri 258 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝑧 ∈ (V × V)) |
| 9 | 8 | adantr 480 | . . . . 5 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝑧 ∈ (V × V)) |
| 10 | 9 | exlimivv 1932 | . . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝑧 ∈ (V × V)) |
| 11 | 10 | abssi 4070 | . . 3 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} ⊆ (V × V) |
| 12 | 3, 11 | eqsstri 4030 | . 2 ⊢ 𝐴 ⊆ (V × V) |
| 13 | df-rel 5692 | . 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
| 14 | 12, 13 | mpbir 231 | 1 ⊢ Rel 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 Vcvv 3480 ⊆ wss 3951 〈cop 4632 {copab 5205 × cxp 5683 Rel wrel 5690 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: (None) |
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