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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 8078 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓)} | |
| 2 | fvex 6919 | . . . . 5 ⊢ (1st ‘𝑤) ∈ V | |
| 3 | fvex 6919 | . . . . 5 ⊢ (2nd ‘𝑤) ∈ V | |
| 4 | eqcom 2744 | . . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑤) ↔ (1st ‘𝑤) = 𝑥) | |
| 5 | eqcom 2744 | . . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑤) ↔ (2nd ‘𝑤) = 𝑦) | |
| 6 | 4, 5 | anbi12i 628 | . . . . . . . . 9 ⊢ ((𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) ↔ ((1st ‘𝑤) = 𝑥 ∧ (2nd ‘𝑤) = 𝑦)) |
| 7 | eqopi 8050 | . . . . . . . . 9 ⊢ ((𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) = 𝑥 ∧ (2nd ‘𝑤) = 𝑦)) → 𝑤 = 〈𝑥, 𝑦〉) | |
| 8 | 6, 7 | sylan2b 594 | . . . . . . . 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 223 | . . . . . 6 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → (𝜓 ↔ 𝜑)) |
| 12 | 11 | ex 412 | . . . . 5 ⊢ (𝑤 ∈ (V × V) → ((𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → (𝜓 ↔ 𝜑))) |
| 13 | 2, 3, 12 | sbc2iedv 3867 | . . . 4 ⊢ (𝑤 ∈ (V × V) → ([(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓 ↔ 𝜑)) |
| 14 | 13 | pm5.32i 574 | . . 3 ⊢ ((𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓) ↔ (𝑤 ∈ (V × V) ∧ 𝜑)) |
| 15 | 14 | opabbii 5210 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓)} = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} |
| 16 | 1, 15 | eqtr2i 2766 | 1 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 [wsbc 3788 〈cop 4632 {copab 5205 × cxp 5683 ‘cfv 6561 {coprab 7432 1st c1st 8012 2nd c2nd 8013 |
| 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: dfoprab4 8080 cnvoprab 8085 df1st2 8123 df2nd2 8124 |
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