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Mirrors > Home > MPE Home > Th. List > ovmpodv | Structured version Visualization version GIF version |
Description: Alternate deduction version of ovmpo 7585, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) |
Ref | Expression |
---|---|
ovmpodf.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
ovmpodf.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷) |
ovmpodf.3 | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉) |
ovmpodf.4 | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅 → 𝜓)) |
Ref | Expression |
---|---|
ovmpodv | ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovmpodf.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
2 | ovmpodf.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷) | |
3 | ovmpodf.3 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉) | |
4 | ovmpodf.4 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅 → 𝜓)) | |
5 | nfcv 2898 | . 2 ⊢ Ⅎ𝑥𝐹 | |
6 | nfv 1909 | . 2 ⊢ Ⅎ𝑥𝜓 | |
7 | nfcv 2898 | . 2 ⊢ Ⅎ𝑦𝐹 | |
8 | nfv 1909 | . 2 ⊢ Ⅎ𝑦𝜓 | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | ovmpodf 7581 | 1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 (class class class)co 7424 ∈ cmpo 7426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-iota 6503 df-fun 6553 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 |
This theorem is referenced by: xpcco 18179 curf12 18224 curf2 18226 |
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