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Mirrors > Home > ILE Home > Th. List > ov2gf | GIF version |
Description: The value of an operation class abstraction. A version of ovmpog 6011 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
ov2gf.a | ⊢ Ⅎ𝑥𝐴 |
ov2gf.c | ⊢ Ⅎ𝑦𝐴 |
ov2gf.d | ⊢ Ⅎ𝑦𝐵 |
ov2gf.1 | ⊢ Ⅎ𝑥𝐺 |
ov2gf.2 | ⊢ Ⅎ𝑦𝑆 |
ov2gf.3 | ⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺) |
ov2gf.4 | ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆) |
ov2gf.5 | ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) |
Ref | Expression |
---|---|
ov2gf | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2750 | . . 3 ⊢ (𝑆 ∈ 𝐻 → 𝑆 ∈ V) | |
2 | ov2gf.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | ov2gf.c | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
4 | ov2gf.d | . . . 4 ⊢ Ⅎ𝑦𝐵 | |
5 | ov2gf.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐺 | |
6 | 5 | nfel1 2330 | . . . . 5 ⊢ Ⅎ𝑥 𝐺 ∈ V |
7 | ov2gf.5 | . . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
8 | nfmpo1 5944 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
9 | 7, 8 | nfcxfr 2316 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 |
10 | nfcv 2319 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
11 | 2, 9, 10 | nfov 5907 | . . . . . 6 ⊢ Ⅎ𝑥(𝐴𝐹𝑦) |
12 | 11, 5 | nfeq 2327 | . . . . 5 ⊢ Ⅎ𝑥(𝐴𝐹𝑦) = 𝐺 |
13 | 6, 12 | nfim 1572 | . . . 4 ⊢ Ⅎ𝑥(𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺) |
14 | ov2gf.2 | . . . . . 6 ⊢ Ⅎ𝑦𝑆 | |
15 | 14 | nfel1 2330 | . . . . 5 ⊢ Ⅎ𝑦 𝑆 ∈ V |
16 | nfmpo2 5945 | . . . . . . . 8 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
17 | 7, 16 | nfcxfr 2316 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 |
18 | 3, 17, 4 | nfov 5907 | . . . . . 6 ⊢ Ⅎ𝑦(𝐴𝐹𝐵) |
19 | 18, 14 | nfeq 2327 | . . . . 5 ⊢ Ⅎ𝑦(𝐴𝐹𝐵) = 𝑆 |
20 | 15, 19 | nfim 1572 | . . . 4 ⊢ Ⅎ𝑦(𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆) |
21 | ov2gf.3 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺) | |
22 | 21 | eleq1d 2246 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑅 ∈ V ↔ 𝐺 ∈ V)) |
23 | oveq1 5884 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦)) | |
24 | 23, 21 | eqeq12d 2192 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = 𝑅 ↔ (𝐴𝐹𝑦) = 𝐺)) |
25 | 22, 24 | imbi12d 234 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅) ↔ (𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺))) |
26 | ov2gf.4 | . . . . . 6 ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆) | |
27 | 26 | eleq1d 2246 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐺 ∈ V ↔ 𝑆 ∈ V)) |
28 | oveq2 5885 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵)) | |
29 | 28, 26 | eqeq12d 2192 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = 𝐺 ↔ (𝐴𝐹𝐵) = 𝑆)) |
30 | 27, 29 | imbi12d 234 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺) ↔ (𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆))) |
31 | 7 | ovmpt4g 5999 | . . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅) |
32 | 31 | 3expia 1205 | . . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅)) |
33 | 2, 3, 4, 13, 20, 25, 30, 32 | vtocl2gaf 2806 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆)) |
34 | 1, 33 | syl5 32 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝑆 ∈ 𝐻 → (𝐴𝐹𝐵) = 𝑆)) |
35 | 34 | 3impia 1200 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 Ⅎwnfc 2306 Vcvv 2739 (class class class)co 5877 ∈ cmpo 5879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-setind 4538 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 |
This theorem is referenced by: (None) |
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