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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 7771 | . 2 ⊢ (𝜑 → (𝐵 × 𝐶) ∈ V) | 
| 5 | wdom2d2.o | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋) | |
| 6 | nfcsb1v 3923 | . . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 | |
| 7 | 6 | nfeq2 2923 | . . . 4 ⊢ Ⅎ𝑦 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 | 
| 8 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑧(1st ‘𝑤) | |
| 9 | nfcsb1v 3923 | . . . . . 6 ⊢ Ⅎ𝑧⦋(2nd ‘𝑤) / 𝑧⦌𝑋 | |
| 10 | 8, 9 | nfcsbw 3925 | . . . . 5 ⊢ Ⅎ𝑧⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 | 
| 11 | 10 | nfeq2 2923 | . . . 4 ⊢ Ⅎ𝑧 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 | 
| 12 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑤 𝑥 = 𝑋 | |
| 13 | csbopeq1a 8075 | . . . . 5 ⊢ (𝑤 = 〈𝑦, 𝑧〉 → ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 = 𝑋) | |
| 14 | 13 | eqeq2d 2748 | . . . 4 ⊢ (𝑤 = 〈𝑦, 𝑧〉 → (𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 ↔ 𝑥 = 𝑋)) | 
| 15 | 7, 11, 12, 14 | rexxpf 5858 | . . 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 9620 | 1 ⊢ (𝜑 → 𝐴 ≼* (𝐵 × 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ⦋csb 3899 〈cop 4632 class class class wbr 5143 × cxp 5683 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 ≼* cwdom 9604 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1st 8014 df-2nd 8015 df-en 8986 df-dom 8987 df-sdom 8988 df-wdom 9605 | 
| This theorem is referenced by: (None) | 
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