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Mirrors > Home > MPE Home > Th. List > relopabiALT | Structured version Visualization version GIF version |
Description: Alternate proof of relopabi 5835 (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 5211 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
3 | 1, 2 | eqtri 2763 | . . 3 ⊢ 𝐴 = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
4 | vex 3482 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | vex 3482 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | opelvv 5729 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ (V × V) |
7 | eleq1 2827 | . . . . . . 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 1930 | . . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝑧 ∈ (V × V)) |
11 | 10 | abssi 4080 | . . 3 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} ⊆ (V × V) |
12 | 3, 11 | eqsstri 4030 | . 2 ⊢ 𝐴 ⊆ (V × V) |
13 | df-rel 5696 | . 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 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 Vcvv 3478 ⊆ wss 3963 〈cop 4637 {copab 5210 × cxp 5687 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-xp 5695 df-rel 5696 |
This theorem is referenced by: (None) |
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