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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 7744 | . . 3 ⊢ {〈𝑎, 𝑧〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
3 | 2 | cnveqi 5738 | . 2 ⊢ ◡{〈𝑎, 𝑧〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
4 | cnvopab 5990 | . . 3 ⊢ ◡{〈𝑎, 𝑧〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {〈𝑧, 𝑎〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} | |
5 | inopab 5694 | . . 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 5428 | . . . 4 ⊢ {〈𝑧, 𝑎〉 ∣ 𝜓} ⊆ {〈𝑧, 𝑎〉 ∣ 𝑎 ∈ (V × V)} |
8 | sseqin2 4190 | . . . 4 ⊢ ({〈𝑧, 𝑎〉 ∣ 𝜓} ⊆ {〈𝑧, 𝑎〉 ∣ 𝑎 ∈ (V × V)} ↔ ({〈𝑧, 𝑎〉 ∣ 𝑎 ∈ (V × V)} ∩ {〈𝑧, 𝑎〉 ∣ 𝜓}) = {〈𝑧, 𝑎〉 ∣ 𝜓}) | |
9 | 7, 8 | mpbi 232 | . . 3 ⊢ ({〈𝑧, 𝑎〉 ∣ 𝑎 ∈ (V × V)} ∩ {〈𝑧, 𝑎〉 ∣ 𝜓}) = {〈𝑧, 𝑎〉 ∣ 𝜓} |
10 | 4, 5, 9 | 3eqtr2i 2848 | . 2 ⊢ ◡{〈𝑎, 𝑧〉 ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {〈𝑧, 𝑎〉 ∣ 𝜓} |
11 | 3, 10 | eqtr3i 2844 | 1 ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑧, 𝑎〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 Vcvv 3493 ∩ cin 3933 ⊆ wss 3934 〈cop 4565 {copab 5119 × cxp 5546 ◡ccnv 5547 {coprab 7149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-oprab 7152 df-1st 7681 df-2nd 7682 |
This theorem is referenced by: f1od2 30449 dfxrn2 35620 |
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