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| Mirrors > Home > MPE Home > Th. List > brfvopabrbr | Structured version Visualization version GIF version | ||
| Description: The binary relation of a function value which is an ordered-pair class abstraction of a restricted binary relation is the restricted binary relation. The first hypothesis can often be obtained by using fvmptopab 7487. (Contributed by AV, 29-Oct-2021.) |
| Ref | Expression |
|---|---|
| brfvopabrbr.1 | ⊢ (𝐴‘𝑍) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝐵‘𝑍)𝑦 ∧ 𝜑)} |
| brfvopabrbr.2 | ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓)) |
| brfvopabrbr.3 | ⊢ Rel (𝐵‘𝑍) |
| Ref | Expression |
|---|---|
| brfvopabrbr | ⊢ (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brne0 5193 | . . . 4 ⊢ (𝑋(𝐴‘𝑍)𝑌 → (𝐴‘𝑍) ≠ ∅) | |
| 2 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐴‘𝑍) = ∅) | |
| 3 | 2 | necon1ai 2968 | . . . 4 ⊢ ((𝐴‘𝑍) ≠ ∅ → 𝑍 ∈ V) |
| 4 | 1, 3 | syl 17 | . . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑍 ∈ V) |
| 5 | brfvopabrbr.1 | . . . . 5 ⊢ (𝐴‘𝑍) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝐵‘𝑍)𝑦 ∧ 𝜑)} | |
| 6 | 5 | relopabiv 5830 | . . . 4 ⊢ Rel (𝐴‘𝑍) |
| 7 | 6 | brrelex1i 5741 | . . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑋 ∈ V) |
| 8 | 6 | brrelex2i 5742 | . . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑌 ∈ V) |
| 9 | 4, 7, 8 | 3jca 1129 | . 2 ⊢ (𝑋(𝐴‘𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V)) |
| 10 | brne0 5193 | . . . . 5 ⊢ (𝑋(𝐵‘𝑍)𝑌 → (𝐵‘𝑍) ≠ ∅) | |
| 11 | fvprc 6898 | . . . . . 6 ⊢ (¬ 𝑍 ∈ V → (𝐵‘𝑍) = ∅) | |
| 12 | 11 | necon1ai 2968 | . . . . 5 ⊢ ((𝐵‘𝑍) ≠ ∅ → 𝑍 ∈ V) |
| 13 | 10, 12 | syl 17 | . . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑍 ∈ V) |
| 14 | brfvopabrbr.3 | . . . . 5 ⊢ Rel (𝐵‘𝑍) | |
| 15 | 14 | brrelex1i 5741 | . . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑋 ∈ V) |
| 16 | 14 | brrelex2i 5742 | . . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑌 ∈ V) |
| 17 | 13, 15, 16 | 3jca 1129 | . . 3 ⊢ (𝑋(𝐵‘𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V)) |
| 18 | 17 | adantr 480 | . 2 ⊢ ((𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓) → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V)) |
| 19 | 5 | a1i 11 | . . 3 ⊢ (𝑍 ∈ V → (𝐴‘𝑍) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝐵‘𝑍)𝑦 ∧ 𝜑)}) |
| 20 | brfvopabrbr.2 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓)) | |
| 21 | 19, 20 | rbropap 5570 | . 2 ⊢ ((𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓))) |
| 22 | 9, 18, 21 | pm5.21nii 378 | 1 ⊢ (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 class class class wbr 5143 {copab 5205 Rel wrel 5690 ‘cfv 6561 |
| 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 |
| 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-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 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-xp 5691 df-rel 5692 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: istrl 29714 ispth 29741 isspth 29742 isclwlk 29793 iscrct 29810 iscycl 29811 iseupth 30220 |
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