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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 7734 | . . . 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 416 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)) |
6 | 5 | alrimiv 1935 | . . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)) |
7 | ssab 3975 | . . . 4 ⊢ (𝐴 ⊆ {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋} ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)) | |
8 | 6, 7 | sylibr 237 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋}) |
9 | 3, 8 | ssexd 5217 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
10 | 9, 1, 4 | wdom2d 9196 | 1 ⊢ (𝜑 → 𝐴 ≼* 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1541 = wceq 1543 ∈ wcel 2110 {cab 2714 ∃wrex 3062 Vcvv 3408 ⊆ wss 3866 class class class wbr 5053 ≼* cwdom 9180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-wdom 9181 |
This theorem is referenced by: hsmexlem2 10041 unxpwdom3 40623 |
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