![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fvopab5 | Structured version Visualization version GIF version |
Description: The value of a function that is expressed as an ordered pair abstraction. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
fvopab5.1 | ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
fvopab5.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
fvopab5 | ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3482 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | df-fv 6562 | . . . 4 ⊢ (𝐹‘𝐴) = (℩𝑧𝐴𝐹𝑧) | |
3 | breq2 5157 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧 ↔ 𝐴𝐹𝑦)) | |
4 | nfcv 2892 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
5 | fvopab5.1 | . . . . . . 7 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
6 | nfopab2 5224 | . . . . . . 7 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
7 | 5, 6 | nfcxfr 2890 | . . . . . 6 ⊢ Ⅎ𝑦𝐹 |
8 | nfcv 2892 | . . . . . 6 ⊢ Ⅎ𝑦𝑧 | |
9 | 4, 7, 8 | nfbr 5200 | . . . . 5 ⊢ Ⅎ𝑦 𝐴𝐹𝑧 |
10 | nfv 1910 | . . . . 5 ⊢ Ⅎ𝑧 𝐴𝐹𝑦 | |
11 | 3, 9, 10 | cbviotaw 6513 | . . . 4 ⊢ (℩𝑧𝐴𝐹𝑧) = (℩𝑦𝐴𝐹𝑦) |
12 | 2, 11 | eqtri 2754 | . . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦) |
13 | nfcv 2892 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
14 | nfopab1 5223 | . . . . . . . 8 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
15 | 5, 14 | nfcxfr 2890 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 |
16 | nfcv 2892 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
17 | 13, 15, 16 | nfbr 5200 | . . . . . 6 ⊢ Ⅎ𝑥 𝐴𝐹𝑦 |
18 | nfv 1910 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
19 | 17, 18 | nfbi 1899 | . . . . 5 ⊢ Ⅎ𝑥(𝐴𝐹𝑦 ↔ 𝜓) |
20 | breq1 5156 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) | |
21 | fvopab5.2 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
22 | 20, 21 | bibi12d 344 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦 ↔ 𝜑) ↔ (𝐴𝐹𝑦 ↔ 𝜓))) |
23 | df-br 5154 | . . . . . 6 ⊢ (𝑥𝐹𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹) | |
24 | 5 | eleq2i 2818 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐹 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
25 | opabidw 5530 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
26 | 23, 24, 25 | 3bitri 296 | . . . . 5 ⊢ (𝑥𝐹𝑦 ↔ 𝜑) |
27 | 19, 22, 26 | vtoclg1f 3551 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑦 ↔ 𝜓)) |
28 | 27 | iotabidv 6538 | . . 3 ⊢ (𝐴 ∈ V → (℩𝑦𝐴𝐹𝑦) = (℩𝑦𝜓)) |
29 | 12, 28 | eqtrid 2778 | . 2 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑦𝜓)) |
30 | 1, 29 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 Vcvv 3462 〈cop 4639 class class class wbr 5153 {copab 5215 ℩cio 6504 ‘cfv 6554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-iota 6506 df-fv 6562 |
This theorem is referenced by: ajval 30794 adjval 31823 |
Copyright terms: Public domain | W3C validator |