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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 3430 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | df-fv 6132 | . . . 4 ⊢ (𝐹‘𝐴) = (℩𝑧𝐴𝐹𝑧) | |
3 | breq2 4878 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧 ↔ 𝐴𝐹𝑦)) | |
4 | nfcv 2970 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
5 | fvopab5.1 | . . . . . . 7 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
6 | nfopab2 4944 | . . . . . . 7 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
7 | 5, 6 | nfcxfr 2968 | . . . . . 6 ⊢ Ⅎ𝑦𝐹 |
8 | nfcv 2970 | . . . . . 6 ⊢ Ⅎ𝑦𝑧 | |
9 | 4, 7, 8 | nfbr 4921 | . . . . 5 ⊢ Ⅎ𝑦 𝐴𝐹𝑧 |
10 | nfv 2015 | . . . . 5 ⊢ Ⅎ𝑧 𝐴𝐹𝑦 | |
11 | 3, 9, 10 | cbviota 6092 | . . . 4 ⊢ (℩𝑧𝐴𝐹𝑧) = (℩𝑦𝐴𝐹𝑦) |
12 | 2, 11 | eqtri 2850 | . . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦) |
13 | nfcv 2970 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
14 | nfopab1 4943 | . . . . . . . 8 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
15 | 5, 14 | nfcxfr 2968 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 |
16 | nfcv 2970 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
17 | 13, 15, 16 | nfbr 4921 | . . . . . 6 ⊢ Ⅎ𝑥 𝐴𝐹𝑦 |
18 | nfv 2015 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
19 | 17, 18 | nfbi 2008 | . . . . 5 ⊢ Ⅎ𝑥(𝐴𝐹𝑦 ↔ 𝜓) |
20 | breq1 4877 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) | |
21 | fvopab5.2 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
22 | 20, 21 | bibi12d 337 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦 ↔ 𝜑) ↔ (𝐴𝐹𝑦 ↔ 𝜓))) |
23 | df-br 4875 | . . . . . 6 ⊢ (𝑥𝐹𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹) | |
24 | 5 | eleq2i 2899 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐹 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
25 | opabid 5209 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
26 | 23, 24, 25 | 3bitri 289 | . . . . 5 ⊢ (𝑥𝐹𝑦 ↔ 𝜑) |
27 | 19, 22, 26 | vtoclg1f 3482 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑦 ↔ 𝜓)) |
28 | 27 | iotabidv 6108 | . . 3 ⊢ (𝐴 ∈ V → (℩𝑦𝐴𝐹𝑦) = (℩𝑦𝜓)) |
29 | 12, 28 | syl5eq 2874 | . 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 198 = wceq 1658 ∈ wcel 2166 Vcvv 3415 〈cop 4404 class class class wbr 4874 {copab 4936 ℩cio 6085 ‘cfv 6124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-iota 6087 df-fv 6132 |
This theorem is referenced by: ajval 28273 adjval 29305 |
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