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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 7940 | . . . 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 1922 | . . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)) |
7 | ssab 4050 | . . . 4 ⊢ (𝐴 ⊆ {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋} ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)) | |
8 | 6, 7 | sylibr 233 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋}) |
9 | 3, 8 | ssexd 5314 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
10 | 9, 1, 4 | wdom2d 9571 | 1 ⊢ (𝜑 → 𝐴 ≼* 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1531 = wceq 1533 ∈ wcel 2098 {cab 2701 ∃wrex 3062 Vcvv 3466 ⊆ wss 3940 class class class wbr 5138 ≼* cwdom 9555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-en 8936 df-dom 8937 df-sdom 8938 df-wdom 9556 |
This theorem is referenced by: hsmexlem2 10418 unxpwdom3 42326 |
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