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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 7656 | . . . 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 415 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)) |
6 | 5 | alrimiv 1924 | . . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)) |
7 | ssab 4041 | . . . 4 ⊢ (𝐴 ⊆ {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋} ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)) | |
8 | 6, 7 | sylibr 236 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋}) |
9 | 3, 8 | ssexd 5221 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
10 | 9, 1, 4 | wdom2d 9038 | 1 ⊢ (𝜑 → 𝐴 ≼* 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1531 = wceq 1533 ∈ wcel 2110 {cab 2799 ∃wrex 3139 Vcvv 3495 ⊆ wss 3936 class class class wbr 5059 ≼* cwdom 9015 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-wdom 9017 |
This theorem is referenced by: hsmexlem2 9843 unxpwdom3 39688 |
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