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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wdom2d2 | Structured version Visualization version GIF version |
Description: Deduction for weak dominance by a Cartesian product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) |
Ref | Expression |
---|---|
wdom2d2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
wdom2d2.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
wdom2d2.c | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
wdom2d2.o | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋) |
Ref | Expression |
---|---|
wdom2d2 | ⊢ (𝜑 → 𝐴 ≼* (𝐵 × 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wdom2d2.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | wdom2d2.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
3 | wdom2d2.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
4 | 2, 3 | xpexd 7770 | . 2 ⊢ (𝜑 → (𝐵 × 𝐶) ∈ V) |
5 | wdom2d2.o | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋) | |
6 | nfcsb1v 3933 | . . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 | |
7 | 6 | nfeq2 2921 | . . . 4 ⊢ Ⅎ𝑦 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 |
8 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑧(1st ‘𝑤) | |
9 | nfcsb1v 3933 | . . . . . 6 ⊢ Ⅎ𝑧⦋(2nd ‘𝑤) / 𝑧⦌𝑋 | |
10 | 8, 9 | nfcsbw 3935 | . . . . 5 ⊢ Ⅎ𝑧⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 |
11 | 10 | nfeq2 2921 | . . . 4 ⊢ Ⅎ𝑧 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 |
12 | nfv 1912 | . . . 4 ⊢ Ⅎ𝑤 𝑥 = 𝑋 | |
13 | csbopeq1a 8074 | . . . . 5 ⊢ (𝑤 = 〈𝑦, 𝑧〉 → ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 = 𝑋) | |
14 | 13 | eqeq2d 2746 | . . . 4 ⊢ (𝑤 = 〈𝑦, 𝑧〉 → (𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 ↔ 𝑥 = 𝑋)) |
15 | 7, 11, 12, 14 | rexxpf 5861 | . . 3 ⊢ (∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋) |
16 | 5, 15 | sylibr 234 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋) |
17 | 1, 4, 16 | wdom2d 9618 | 1 ⊢ (𝜑 → 𝐴 ≼* (𝐵 × 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ⦋csb 3908 〈cop 4637 class class class wbr 5148 × cxp 5687 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 ≼* cwdom 9602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-1st 8013 df-2nd 8014 df-en 8985 df-dom 8986 df-sdom 8987 df-wdom 9603 |
This theorem is referenced by: (None) |
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