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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 7742 | . 2 ⊢ (𝜑 → (𝐵 × 𝐶) ∈ V) |
5 | wdom2d2.o | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋) | |
6 | nfcsb1v 3918 | . . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 | |
7 | 6 | nfeq2 2919 | . . . 4 ⊢ Ⅎ𝑦 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 |
8 | nfcv 2902 | . . . . . 6 ⊢ Ⅎ𝑧(1st ‘𝑤) | |
9 | nfcsb1v 3918 | . . . . . 6 ⊢ Ⅎ𝑧⦋(2nd ‘𝑤) / 𝑧⦌𝑋 | |
10 | 8, 9 | nfcsbw 3920 | . . . . 5 ⊢ Ⅎ𝑧⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 |
11 | 10 | nfeq2 2919 | . . . 4 ⊢ Ⅎ𝑧 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 |
12 | nfv 1916 | . . . 4 ⊢ Ⅎ𝑤 𝑥 = 𝑋 | |
13 | csbopeq1a 8040 | . . . . 5 ⊢ (𝑤 = 〈𝑦, 𝑧〉 → ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 = 𝑋) | |
14 | 13 | eqeq2d 2742 | . . . 4 ⊢ (𝑤 = 〈𝑦, 𝑧〉 → (𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 ↔ 𝑥 = 𝑋)) |
15 | 7, 11, 12, 14 | rexxpf 5847 | . . 3 ⊢ (∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋) |
16 | 5, 15 | sylibr 233 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋) |
17 | 1, 4, 16 | wdom2d 9581 | 1 ⊢ (𝜑 → 𝐴 ≼* (𝐵 × 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 Vcvv 3473 ⦋csb 3893 〈cop 4634 class class class wbr 5148 × cxp 5674 ‘cfv 6543 1st c1st 7977 2nd c2nd 7978 ≼* cwdom 9565 |
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 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-1st 7979 df-2nd 7980 df-en 8946 df-dom 8947 df-sdom 8948 df-wdom 9566 |
This theorem is referenced by: (None) |
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