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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 7477 | . 2 ⊢ (𝜑 → (𝐵 × 𝐶) ∈ V) |
5 | wdom2d2.o | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋) | |
6 | nfcsb1v 3910 | . . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 | |
7 | 6 | nfeq2 2998 | . . . 4 ⊢ Ⅎ𝑦 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 |
8 | nfcv 2980 | . . . . . 6 ⊢ Ⅎ𝑧(1st ‘𝑤) | |
9 | nfcsb1v 3910 | . . . . . 6 ⊢ Ⅎ𝑧⦋(2nd ‘𝑤) / 𝑧⦌𝑋 | |
10 | 8, 9 | nfcsbw 3912 | . . . . 5 ⊢ Ⅎ𝑧⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 |
11 | 10 | nfeq2 2998 | . . . 4 ⊢ Ⅎ𝑧 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 |
12 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑤 𝑥 = 𝑋 | |
13 | csbopeq1a 7752 | . . . . 5 ⊢ (𝑤 = 〈𝑦, 𝑧〉 → ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 = 𝑋) | |
14 | 13 | eqeq2d 2835 | . . . 4 ⊢ (𝑤 = 〈𝑦, 𝑧〉 → (𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 ↔ 𝑥 = 𝑋)) |
15 | 7, 11, 12, 14 | rexxpf 5721 | . . 3 ⊢ (∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋) |
16 | 5, 15 | sylibr 236 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋) |
17 | 1, 4, 16 | wdom2d 9047 | 1 ⊢ (𝜑 → 𝐴 ≼* (𝐵 × 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 Vcvv 3497 ⦋csb 3886 〈cop 4576 class class class wbr 5069 × cxp 5556 ‘cfv 6358 1st c1st 7690 2nd c2nd 7691 ≼* cwdom 9024 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-1st 7692 df-2nd 7693 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-wdom 9026 |
This theorem is referenced by: (None) |
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