![]() |
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 3466 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | df-fv 6509 | . . . 4 ⊢ (𝐹‘𝐴) = (℩𝑧𝐴𝐹𝑧) | |
3 | breq2 5114 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧 ↔ 𝐴𝐹𝑦)) | |
4 | nfcv 2908 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
5 | fvopab5.1 | . . . . . . 7 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
6 | nfopab2 5181 | . . . . . . 7 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
7 | 5, 6 | nfcxfr 2906 | . . . . . 6 ⊢ Ⅎ𝑦𝐹 |
8 | nfcv 2908 | . . . . . 6 ⊢ Ⅎ𝑦𝑧 | |
9 | 4, 7, 8 | nfbr 5157 | . . . . 5 ⊢ Ⅎ𝑦 𝐴𝐹𝑧 |
10 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑧 𝐴𝐹𝑦 | |
11 | 3, 9, 10 | cbviotaw 6460 | . . . 4 ⊢ (℩𝑧𝐴𝐹𝑧) = (℩𝑦𝐴𝐹𝑦) |
12 | 2, 11 | eqtri 2765 | . . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦) |
13 | nfcv 2908 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
14 | nfopab1 5180 | . . . . . . . 8 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
15 | 5, 14 | nfcxfr 2906 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 |
16 | nfcv 2908 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
17 | 13, 15, 16 | nfbr 5157 | . . . . . 6 ⊢ Ⅎ𝑥 𝐴𝐹𝑦 |
18 | nfv 1918 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
19 | 17, 18 | nfbi 1907 | . . . . 5 ⊢ Ⅎ𝑥(𝐴𝐹𝑦 ↔ 𝜓) |
20 | breq1 5113 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) | |
21 | fvopab5.2 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
22 | 20, 21 | bibi12d 346 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦 ↔ 𝜑) ↔ (𝐴𝐹𝑦 ↔ 𝜓))) |
23 | df-br 5111 | . . . . . 6 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹) | |
24 | 5 | eleq2i 2830 | . . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) |
25 | opabidw 5486 | . . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑) | |
26 | 23, 24, 25 | 3bitri 297 | . . . . 5 ⊢ (𝑥𝐹𝑦 ↔ 𝜑) |
27 | 19, 22, 26 | vtoclg1f 3527 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑦 ↔ 𝜓)) |
28 | 27 | iotabidv 6485 | . . 3 ⊢ (𝐴 ∈ V → (℩𝑦𝐴𝐹𝑦) = (℩𝑦𝜓)) |
29 | 12, 28 | eqtrid 2789 | . 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 1542 ∈ wcel 2107 Vcvv 3448 ⟨cop 4597 class class class wbr 5110 {copab 5172 ℩cio 6451 ‘cfv 6501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-iota 6453 df-fv 6509 |
This theorem is referenced by: ajval 29845 adjval 30874 |
Copyright terms: Public domain | W3C validator |