Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7198. (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 5107 | . . . 4 ⊢ (𝑋(𝐴‘𝑍)𝑌 → (𝐴‘𝑍) ≠ ∅) | |
2 | fvprc 6656 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐴‘𝑍) = ∅) | |
3 | 2 | necon1ai 3040 | . . . 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 5687 | . . . 4 ⊢ Rel (𝐴‘𝑍) |
7 | 6 | brrelex1i 5601 | . . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑋 ∈ V) |
8 | 6 | brrelex2i 5602 | . . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑌 ∈ V) |
9 | 4, 7, 8 | 3jca 1120 | . 2 ⊢ (𝑋(𝐴‘𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V)) |
10 | brne0 5107 | . . . . 5 ⊢ (𝑋(𝐵‘𝑍)𝑌 → (𝐵‘𝑍) ≠ ∅) | |
11 | fvprc 6656 | . . . . . 6 ⊢ (¬ 𝑍 ∈ V → (𝐵‘𝑍) = ∅) | |
12 | 11 | necon1ai 3040 | . . . . 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 5601 | . . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑋 ∈ V) |
16 | 14 | brrelex2i 5602 | . . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑌 ∈ V) |
17 | 13, 15, 16 | 3jca 1120 | . . 3 ⊢ (𝑋(𝐵‘𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V)) |
18 | 17 | adantr 481 | . 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 5441 | . 2 ⊢ ((𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓))) |
22 | 9, 18, 21 | pm5.21nii 380 | 1 ⊢ (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ∅c0 4288 class class class wbr 5057 {copab 5119 Rel wrel 5553 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-iota 6307 df-fv 6356 |
This theorem is referenced by: istrl 27405 ispth 27431 isspth 27432 isclwlk 27481 iscrct 27498 iscycl 27499 iseupth 27907 |
Copyright terms: Public domain | W3C validator |