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| Mirrors > Home > MPE Home > Th. List > brfvopab | Structured version Visualization version GIF version | ||
| Description: The classes involved in a binary relation of a function value which is an ordered-pair class abstraction are sets. (Contributed by AV, 7-Jan-2021.) |
| Ref | Expression |
|---|---|
| brfvopab.1 | ⊢ (𝑋 ∈ V → (𝐹‘𝑋) = {〈𝑦, 𝑧〉 ∣ 𝜑}) |
| Ref | Expression |
|---|---|
| brfvopab | ⊢ (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brfvopab.1 | . . . . . . 7 ⊢ (𝑋 ∈ V → (𝐹‘𝑋) = {〈𝑦, 𝑧〉 ∣ 𝜑}) | |
| 2 | 1 | breqd 5154 | . . . . . 6 ⊢ (𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 ↔ 𝐴{〈𝑦, 𝑧〉 ∣ 𝜑}𝐵)) |
| 3 | brabv 5573 | . . . . . 6 ⊢ (𝐴{〈𝑦, 𝑧〉 ∣ 𝜑}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
| 4 | 2, 3 | biimtrdi 253 | . . . . 5 ⊢ (𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
| 5 | 4 | imdistani 568 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝐴(𝐹‘𝑋)𝐵) → (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
| 6 | 3anass 1095 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V))) | |
| 7 | 5, 6 | sylibr 234 | . . 3 ⊢ ((𝑋 ∈ V ∧ 𝐴(𝐹‘𝑋)𝐵) → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)) |
| 8 | 7 | ex 412 | . 2 ⊢ (𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))) |
| 9 | fvprc 6898 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
| 10 | breq 5145 | . . . 4 ⊢ ((𝐹‘𝑋) = ∅ → (𝐴(𝐹‘𝑋)𝐵 ↔ 𝐴∅𝐵)) | |
| 11 | br0 5192 | . . . . 5 ⊢ ¬ 𝐴∅𝐵 | |
| 12 | 11 | pm2.21i 119 | . . . 4 ⊢ (𝐴∅𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)) |
| 13 | 10, 12 | biimtrdi 253 | . . 3 ⊢ ((𝐹‘𝑋) = ∅ → (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))) |
| 14 | 9, 13 | syl 17 | . 2 ⊢ (¬ 𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))) |
| 15 | 8, 14 | pm2.61i 182 | 1 ⊢ (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 class class class wbr 5143 {copab 5205 ‘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-iota 6514 df-fv 6569 |
| This theorem is referenced by: wlkprop 29629 wlkv 29630 isupwlkg 48053 |
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