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| 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 7985 | . . . 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 1927 | . . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)) | 
| 7 | ssab 4064 | . . . 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 5324 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) | 
| 10 | 9, 1, 4 | wdom2d 9620 | 1 ⊢ (𝜑 → 𝐴 ≼* 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ≼* cwdom 9604 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-en 8986 df-dom 8987 df-sdom 8988 df-wdom 9605 | 
| This theorem is referenced by: hsmexlem2 10467 unxpwdom3 43107 | 
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