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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 5079 | . . . . . 6 ⊢ (𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 ↔ 𝐴{〈𝑦, 𝑧〉 ∣ 𝜑}𝐵)) |
3 | brabv 5455 | . . . . . 6 ⊢ (𝐴{〈𝑦, 𝑧〉 ∣ 𝜑}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
4 | 2, 3 | syl6bi 255 | . . . . 5 ⊢ (𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
5 | 4 | imdistani 571 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝐴(𝐹‘𝑋)𝐵) → (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
6 | 3anass 1091 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V))) | |
7 | 5, 6 | sylibr 236 | . . 3 ⊢ ((𝑋 ∈ V ∧ 𝐴(𝐹‘𝑋)𝐵) → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)) |
8 | 7 | ex 415 | . 2 ⊢ (𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))) |
9 | fvprc 6665 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
10 | breq 5070 | . . . 4 ⊢ ((𝐹‘𝑋) = ∅ → (𝐴(𝐹‘𝑋)𝐵 ↔ 𝐴∅𝐵)) | |
11 | br0 5117 | . . . . 5 ⊢ ¬ 𝐴∅𝐵 | |
12 | 11 | pm2.21i 119 | . . . 4 ⊢ (𝐴∅𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)) |
13 | 10, 12 | syl6bi 255 | . . 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 184 | 1 ⊢ (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 class class class wbr 5068 {copab 5130 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-iota 6316 df-fv 6365 |
This theorem is referenced by: wlkprop 27395 wlkv 27396 isupwlkg 44019 |
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