![]() |
Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
|
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 7686 | . 2 ⊢ (𝜑 → (𝐵 × 𝐶) ∈ V) |
5 | wdom2d2.o | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋) | |
6 | nfcsb1v 3881 | . . . . 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 2904 | . . . . . 6 ⊢ Ⅎ𝑧(1st ‘𝑤) | |
9 | nfcsb1v 3881 | . . . . . 6 ⊢ Ⅎ𝑧⦋(2nd ‘𝑤) / 𝑧⦌𝑋 | |
10 | 8, 9 | nfcsbw 3883 | . . . . 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 1918 | . . . 4 ⊢ Ⅎ𝑤 𝑥 = 𝑋 | |
13 | csbopeq1a 7983 | . . . . 5 ⊢ (𝑤 = ⟨𝑦, 𝑧⟩ → ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 = 𝑋) | |
14 | 13 | eqeq2d 2744 | . . . 4 ⊢ (𝑤 = ⟨𝑦, 𝑧⟩ → (𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 ↔ 𝑥 = 𝑋)) |
15 | 7, 11, 12, 14 | rexxpf 5804 | . . 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 9521 | 1 ⊢ (𝜑 → 𝐴 ≼* (𝐵 × 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3070 Vcvv 3444 ⦋csb 3856 ⟨cop 4593 class class class wbr 5106 × cxp 5632 ‘cfv 6497 1st c1st 7920 2nd c2nd 7921 ≼* cwdom 9505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-1st 7922 df-2nd 7923 df-en 8887 df-dom 8888 df-sdom 8889 df-wdom 9506 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |