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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 6844. (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 4836 | . . . 4 ⊢ (𝑋(𝐴‘𝑍)𝑌 → (𝐴‘𝑍) ≠ ∅) | |
2 | fvprc 6326 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐴‘𝑍) = ∅) | |
3 | 2 | necon1ai 2970 | . . . 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 | relopabi 5384 | . . . 4 ⊢ Rel (𝐴‘𝑍) |
7 | 6 | brrelexi 5298 | . . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑋 ∈ V) |
8 | 6 | brrelex2i 5299 | . . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑌 ∈ V) |
9 | 4, 7, 8 | 3jca 1122 | . 2 ⊢ (𝑋(𝐴‘𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V)) |
10 | brne0 4836 | . . . . 5 ⊢ (𝑋(𝐵‘𝑍)𝑌 → (𝐵‘𝑍) ≠ ∅) | |
11 | fvprc 6326 | . . . . . 6 ⊢ (¬ 𝑍 ∈ V → (𝐵‘𝑍) = ∅) | |
12 | 11 | necon1ai 2970 | . . . . 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 | brrelexi 5298 | . . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑋 ∈ V) |
16 | 14 | brrelex2i 5299 | . . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑌 ∈ V) |
17 | 13, 15, 16 | 3jca 1122 | . . 3 ⊢ (𝑋(𝐵‘𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V)) |
18 | 17 | adantr 466 | . 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 5149 | . 2 ⊢ ((𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓))) |
22 | 9, 18, 21 | pm5.21nii 367 | 1 ⊢ (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 Vcvv 3351 ∅c0 4063 class class class wbr 4786 {copab 4846 Rel wrel 5254 ‘cfv 6031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-xp 5255 df-rel 5256 df-iota 5994 df-fv 6039 |
This theorem is referenced by: istrl 26828 ispth 26854 isspth 26855 isclwlk 26904 iscrct 26921 iscycl 26922 iseupth 27381 |
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