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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 3185 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | df-fv 5812 | . . . 4 ⊢ (𝐹‘𝐴) = (℩𝑧𝐴𝐹𝑧) | |
3 | breq2 4587 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧 ↔ 𝐴𝐹𝑦)) | |
4 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
5 | fvopab5.1 | . . . . . . 7 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
6 | nfopab2 4652 | . . . . . . 7 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
7 | 5, 6 | nfcxfr 2749 | . . . . . 6 ⊢ Ⅎ𝑦𝐹 |
8 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑦𝑧 | |
9 | 4, 7, 8 | nfbr 4629 | . . . . 5 ⊢ Ⅎ𝑦 𝐴𝐹𝑧 |
10 | nfv 1830 | . . . . 5 ⊢ Ⅎ𝑧 𝐴𝐹𝑦 | |
11 | 3, 9, 10 | cbviota 5773 | . . . 4 ⊢ (℩𝑧𝐴𝐹𝑧) = (℩𝑦𝐴𝐹𝑦) |
12 | 2, 11 | eqtri 2632 | . . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦) |
13 | nfcv 2751 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
14 | nfopab1 4651 | . . . . . . . 8 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
15 | 5, 14 | nfcxfr 2749 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 |
16 | nfcv 2751 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
17 | 13, 15, 16 | nfbr 4629 | . . . . . 6 ⊢ Ⅎ𝑥 𝐴𝐹𝑦 |
18 | nfv 1830 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
19 | 17, 18 | nfbi 1821 | . . . . 5 ⊢ Ⅎ𝑥(𝐴𝐹𝑦 ↔ 𝜓) |
20 | breq1 4586 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) | |
21 | fvopab5.2 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
22 | 20, 21 | bibi12d 334 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦 ↔ 𝜑) ↔ (𝐴𝐹𝑦 ↔ 𝜓))) |
23 | df-br 4584 | . . . . . 6 ⊢ (𝑥𝐹𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹) | |
24 | 5 | eleq2i 2680 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐹 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
25 | opabid 4907 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
26 | 23, 24, 25 | 3bitri 285 | . . . . 5 ⊢ (𝑥𝐹𝑦 ↔ 𝜑) |
27 | 19, 22, 26 | vtoclg1f 3238 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑦 ↔ 𝜓)) |
28 | 27 | iotabidv 5789 | . . 3 ⊢ (𝐴 ∈ V → (℩𝑦𝐴𝐹𝑦) = (℩𝑦𝜓)) |
29 | 12, 28 | syl5eq 2656 | . 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 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 class class class wbr 4583 {copab 4642 ℩cio 5766 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-iota 5768 df-fv 5812 |
This theorem is referenced by: ajval 27101 adjval 28133 |
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