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Mirrors > Home > MPE Home > Th. List > wdomd | Structured version Visualization version GIF version |
Description: Deduce weak dominance from an implicit onto function. (Contributed by Stefan O'Rear, 13-Feb-2015.) |
Ref | Expression |
---|---|
wdomd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
wdomd.o | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋) |
Ref | Expression |
---|---|
wdomd | ⊢ (𝜑 → 𝐴 ≼* 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wdomd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
2 | abrexexg 7984 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋} ∈ V) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋} ∈ V) |
4 | wdomd.o | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋) | |
5 | 4 | ex 412 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)) |
6 | 5 | alrimiv 1925 | . . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)) |
7 | ssab 4074 | . . . 4 ⊢ (𝐴 ⊆ {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋} ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)) | |
8 | 6, 7 | sylibr 234 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋}) |
9 | 3, 8 | ssexd 5330 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
10 | 9, 1, 4 | wdom2d 9618 | 1 ⊢ (𝜑 → 𝐴 ≼* 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 = wceq 1537 ∈ wcel 2106 {cab 2712 ∃wrex 3068 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 ≼* 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-rep 5285 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-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-en 8985 df-dom 8986 df-sdom 8987 df-wdom 9603 |
This theorem is referenced by: hsmexlem2 10465 unxpwdom3 43084 |
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