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| 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 8079 | . . 3 ⊢ {〈𝑎, 𝑧〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
| 3 | 2 | cnveqi 5885 | . 2 ⊢ ◡{〈𝑎, 𝑧〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
| 4 | cnvopab 6157 | . . 3 ⊢ ◡{〈𝑎, 𝑧〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {〈𝑧, 𝑎〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} | |
| 5 | inopab 5839 | . . 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 5555 | . . . 4 ⊢ {〈𝑧, 𝑎〉 ∣ 𝜓} ⊆ {〈𝑧, 𝑎〉 ∣ 𝑎 ∈ (V × V)} |
| 8 | sseqin2 4223 | . . . 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 2771 | . 2 ⊢ ◡{〈𝑎, 𝑧〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {〈𝑧, 𝑎〉 ∣ 𝜓} |
| 11 | 3, 10 | eqtr3i 2767 | 1 ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑧, 𝑎〉 ∣ 𝜓} |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 〈cop 4632 {copab 5205 × cxp 5683 ◡ccnv 5684 {coprab 7432 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-oprab 7435 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: f1od2 32732 dfxrn2 38377 |
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