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Mirrors > Home > ILE Home > Th. List > ovmpt2dv2 | GIF version |
Description: Alternate deduction version of ovmpt2 5688, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) |
Ref | Expression |
---|---|
ovmpt2dv2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
ovmpt2dv2.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷) |
ovmpt2dv2.3 | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉) |
ovmpt2dv2.4 | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) |
Ref | Expression |
---|---|
ovmpt2dv2 | ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → (𝐴𝐹𝐵) = 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2084 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) | |
2 | ovmpt2dv2.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
3 | ovmpt2dv2.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷) | |
4 | ovmpt2dv2.3 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉) | |
5 | ovmpt2dv2.4 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) | |
6 | 5 | eqeq2d 2094 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑅 ↔ (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)) |
7 | 6 | biimpd 142 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑅 → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)) |
8 | nfmpt21 5623 | . . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
9 | nfcv 2223 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
10 | nfcv 2223 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
11 | 9, 8, 10 | nfov 5587 | . . . . 5 ⊢ Ⅎ𝑥(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) |
12 | 11 | nfeq1 2232 | . . . 4 ⊢ Ⅎ𝑥(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆 |
13 | nfmpt22 5624 | . . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
14 | nfcv 2223 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
15 | nfcv 2223 | . . . . . 6 ⊢ Ⅎ𝑦𝐵 | |
16 | 14, 13, 15 | nfov 5587 | . . . . 5 ⊢ Ⅎ𝑦(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) |
17 | 16 | nfeq1 2232 | . . . 4 ⊢ Ⅎ𝑦(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆 |
18 | 2, 3, 4, 7, 8, 12, 13, 17 | ovmpt2df 5684 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)) |
19 | 1, 18 | mpd 13 | . 2 ⊢ (𝜑 → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆) |
20 | oveq 5570 | . . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → (𝐴𝐹𝐵) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵)) | |
21 | 20 | eqeq1d 2091 | . 2 ⊢ (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → ((𝐴𝐹𝐵) = 𝑆 ↔ (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)) |
22 | 19, 21 | syl5ibrcom 155 | 1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → (𝐴𝐹𝐵) = 𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 (class class class)co 5564 ↦ cmpt2 5566 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3917 ax-pow 3969 ax-pr 3993 ax-setind 4309 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-v 2612 df-sbc 2826 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-br 3807 df-opab 3861 df-id 4077 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-iota 4918 df-fun 4955 df-fv 4961 df-ov 5567 df-oprab 5568 df-mpt2 5569 |
This theorem is referenced by: (None) |
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