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Mirrors > Home > MPE Home > Th. List > brwdom3i | Structured version Visualization version GIF version |
Description: Weak dominance implies existence of a covering function. (Contributed by Stefan O'Rear, 13-Feb-2015.) |
Ref | Expression |
---|---|
brwdom3i | ⊢ (𝑋 ≼* 𝑌 → ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relwdom 8823 | . . . 4 ⊢ Rel ≼* | |
2 | 1 | brrelex1i 5454 | . . 3 ⊢ (𝑋 ≼* 𝑌 → 𝑋 ∈ V) |
3 | 1 | brrelex2i 5455 | . . 3 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
4 | brwdom3 8839 | . . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ≼* 𝑌 ↔ ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦))) | |
5 | 2, 3, 4 | syl2anc 576 | . 2 ⊢ (𝑋 ≼* 𝑌 → (𝑋 ≼* 𝑌 ↔ ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦))) |
6 | 5 | ibi 259 | 1 ⊢ (𝑋 ≼* 𝑌 → ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1508 ∃wex 1743 ∈ wcel 2051 ∀wral 3081 ∃wrex 3082 Vcvv 3408 class class class wbr 4925 ‘cfv 6185 ≼* cwdom 8814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-wdom 8816 |
This theorem is referenced by: unwdomg 8841 xpwdomg 8842 |
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