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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 4594 | . . . . . 6 ⊢ (𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 ↔ 𝐴{〈𝑦, 𝑧〉 ∣ 𝜑}𝐵)) |
3 | brabv 6597 | . . . . . 6 ⊢ (𝐴{〈𝑦, 𝑧〉 ∣ 𝜑}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
4 | 2, 3 | syl6bi 242 | . . . . 5 ⊢ (𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
5 | 4 | imdistani 722 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝐴(𝐹‘𝑋)𝐵) → (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
6 | 3anass 1035 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V))) | |
7 | 5, 6 | sylibr 223 | . . 3 ⊢ ((𝑋 ∈ V ∧ 𝐴(𝐹‘𝑋)𝐵) → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)) |
8 | 7 | ex 449 | . 2 ⊢ (𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))) |
9 | fvprc 6097 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
10 | breq 4585 | . . . 4 ⊢ ((𝐹‘𝑋) = ∅ → (𝐴(𝐹‘𝑋)𝐵 ↔ 𝐴∅𝐵)) | |
11 | br0 4631 | . . . . 5 ⊢ ¬ 𝐴∅𝐵 | |
12 | 11 | pm2.21i 115 | . . . 4 ⊢ (𝐴∅𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)) |
13 | 10, 12 | syl6bi 242 | . . 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 175 | 1 ⊢ (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 class class class wbr 4583 {copab 4642 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-iota 5768 df-fv 5812 |
This theorem is referenced by: wlkv 40815 trlis1wlk 40905 PthisTrl 40931 sPthisPth 40932 spthdep 40940 pthdepissPth 40941 isclWlkb 40980 clwlkis1wlk 40981 crctprop 40998 cyclprop 40999 eupthtrli 41370 eupthi 41371 upgreupthi 41376 |
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