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Mirrors > Home > MPE Home > Th. List > ovmpodv | Structured version Visualization version GIF version |
Description: Alternate deduction version of ovmpo 7495, 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 2904 | . 2 ⊢ Ⅎ𝑥𝐹 | |
6 | nfv 1916 | . 2 ⊢ Ⅎ𝑥𝜓 | |
7 | nfcv 2904 | . 2 ⊢ Ⅎ𝑦𝐹 | |
8 | nfv 1916 | . 2 ⊢ Ⅎ𝑦𝜓 | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | ovmpodf 7491 | 1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 (class class class)co 7337 ∈ cmpo 7339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-iota 6431 df-fun 6481 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 |
This theorem is referenced by: xpcco 17997 curf12 18042 curf2 18044 |
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