![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cnvoprab | Structured version Visualization version GIF version |
Description: The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) (Proof shortened by Thierry Arnoux, 20-Feb-2022.) |
Ref | Expression |
---|---|
cnvoprab.1 | ⊢ (𝑎 = 〈𝑥, 𝑦〉 → (𝜓 ↔ 𝜑)) |
cnvoprab.2 | ⊢ (𝜓 → 𝑎 ∈ (V × V)) |
Ref | Expression |
---|---|
cnvoprab | ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑧, 𝑎〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvoprab.1 | . . . 4 ⊢ (𝑎 = 〈𝑥, 𝑦〉 → (𝜓 ↔ 𝜑)) | |
2 | 1 | dfoprab3 8078 | . . 3 ⊢ {〈𝑎, 𝑧〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
3 | 2 | cnveqi 5888 | . 2 ⊢ ◡{〈𝑎, 𝑧〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
4 | cnvopab 6160 | . . 3 ⊢ ◡{〈𝑎, 𝑧〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {〈𝑧, 𝑎〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} | |
5 | inopab 5842 | . . 3 ⊢ ({〈𝑧, 𝑎〉 ∣ 𝑎 ∈ (V × V)} ∩ {〈𝑧, 𝑎〉 ∣ 𝜓}) = {〈𝑧, 𝑎〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} | |
6 | cnvoprab.2 | . . . . 5 ⊢ (𝜓 → 𝑎 ∈ (V × V)) | |
7 | 6 | ssopab2i 5560 | . . . 4 ⊢ {〈𝑧, 𝑎〉 ∣ 𝜓} ⊆ {〈𝑧, 𝑎〉 ∣ 𝑎 ∈ (V × V)} |
8 | sseqin2 4231 | . . . 4 ⊢ ({〈𝑧, 𝑎〉 ∣ 𝜓} ⊆ {〈𝑧, 𝑎〉 ∣ 𝑎 ∈ (V × V)} ↔ ({〈𝑧, 𝑎〉 ∣ 𝑎 ∈ (V × V)} ∩ {〈𝑧, 𝑎〉 ∣ 𝜓}) = {〈𝑧, 𝑎〉 ∣ 𝜓}) | |
9 | 7, 8 | mpbi 230 | . . 3 ⊢ ({〈𝑧, 𝑎〉 ∣ 𝑎 ∈ (V × V)} ∩ {〈𝑧, 𝑎〉 ∣ 𝜓}) = {〈𝑧, 𝑎〉 ∣ 𝜓} |
10 | 4, 5, 9 | 3eqtr2i 2769 | . 2 ⊢ ◡{〈𝑎, 𝑧〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {〈𝑧, 𝑎〉 ∣ 𝜓} |
11 | 3, 10 | eqtr3i 2765 | 1 ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑧, 𝑎〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 〈cop 4637 {copab 5210 × cxp 5687 ◡ccnv 5688 {coprab 7432 |
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-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
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-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-oprab 7435 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: f1od2 32739 dfxrn2 38358 |
Copyright terms: Public domain | W3C validator |