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Mirrors > Home > MPE Home > Th. List > dfoprab3 | Structured version Visualization version GIF version |
Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.) |
Ref | Expression |
---|---|
dfoprab3.1 | ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
dfoprab3 | ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfoprab3s 7832 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓)} | |
2 | fvex 6739 | . . . . 5 ⊢ (1st ‘𝑤) ∈ V | |
3 | fvex 6739 | . . . . 5 ⊢ (2nd ‘𝑤) ∈ V | |
4 | eqcom 2745 | . . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑤) ↔ (1st ‘𝑤) = 𝑥) | |
5 | eqcom 2745 | . . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑤) ↔ (2nd ‘𝑤) = 𝑦) | |
6 | 4, 5 | anbi12i 630 | . . . . . . . . 9 ⊢ ((𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) ↔ ((1st ‘𝑤) = 𝑥 ∧ (2nd ‘𝑤) = 𝑦)) |
7 | eqopi 7806 | . . . . . . . . 9 ⊢ ((𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) = 𝑥 ∧ (2nd ‘𝑤) = 𝑦)) → 𝑤 = 〈𝑥, 𝑦〉) | |
8 | 6, 7 | sylan2b 597 | . . . . . . . 8 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → 𝑤 = 〈𝑥, 𝑦〉) |
9 | dfoprab3.1 | . . . . . . . 8 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) | |
10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → (𝜑 ↔ 𝜓)) |
11 | 10 | bicomd 226 | . . . . . 6 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → (𝜓 ↔ 𝜑)) |
12 | 11 | ex 416 | . . . . 5 ⊢ (𝑤 ∈ (V × V) → ((𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → (𝜓 ↔ 𝜑))) |
13 | 2, 3, 12 | sbc2iedv 3789 | . . . 4 ⊢ (𝑤 ∈ (V × V) → ([(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓 ↔ 𝜑)) |
14 | 13 | pm5.32i 578 | . . 3 ⊢ ((𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓) ↔ (𝑤 ∈ (V × V) ∧ 𝜑)) |
15 | 14 | opabbii 5129 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓)} = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} |
16 | 1, 15 | eqtr2i 2767 | 1 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2111 Vcvv 3415 [wsbc 3703 〈cop 4556 {copab 5124 × cxp 5558 ‘cfv 6389 {coprab 7223 1st c1st 7768 2nd c2nd 7769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pr 5331 ax-un 7532 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3417 df-sbc 3704 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-nul 4247 df-if 4449 df-sn 4551 df-pr 4553 df-op 4557 df-uni 4829 df-br 5063 df-opab 5125 df-mpt 5145 df-id 5464 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-iota 6347 df-fun 6391 df-fv 6397 df-oprab 7226 df-1st 7770 df-2nd 7771 |
This theorem is referenced by: dfoprab4 7834 cnvoprab 7839 df1st2 7875 df2nd2 7876 |
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