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Mirrors > Home > MPE Home > Th. List > fowdom | Structured version Visualization version GIF version |
Description: An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Ref | Expression |
---|---|
fowdom | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ≼* 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3511 | . 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | |
2 | foeq1 6579 | . . . . . 6 ⊢ (𝑧 = 𝐹 → (𝑧:𝑌–onto→𝑋 ↔ 𝐹:𝑌–onto→𝑋)) | |
3 | 2 | spcegv 3595 | . . . . 5 ⊢ (𝐹 ∈ V → (𝐹:𝑌–onto→𝑋 → ∃𝑧 𝑧:𝑌–onto→𝑋)) |
4 | 3 | imp 409 | . . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → ∃𝑧 𝑧:𝑌–onto→𝑋) |
5 | 4 | olcd 870 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
6 | fof 6583 | . . . . 5 ⊢ (𝐹:𝑌–onto→𝑋 → 𝐹:𝑌⟶𝑋) | |
7 | dmfex 7633 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌⟶𝑋) → 𝑌 ∈ V) | |
8 | 6, 7 | sylan2 594 | . . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → 𝑌 ∈ V) |
9 | brwdom 9023 | . . . 4 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) |
11 | 5, 10 | mpbird 259 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ≼* 𝑌) |
12 | 1, 11 | sylan 582 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ≼* 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1531 ∃wex 1774 ∈ wcel 2108 Vcvv 3493 ∅c0 4289 class class class wbr 5057 ⟶wf 6344 –onto→wfo 6346 ≼* cwdom 9013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-wdom 9015 |
This theorem is referenced by: wdomref 9028 wdomtr 9031 wdom2d 9036 wdomima2g 9042 harwdom 9046 ixpiunwdom 9047 isf32lem10 9776 fin1a2lem7 9820 |
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