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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 3465 | . 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | |
2 | foeq1 6756 | . . . . . 6 ⊢ (𝑧 = 𝐹 → (𝑧:𝑌–onto→𝑋 ↔ 𝐹:𝑌–onto→𝑋)) | |
3 | 2 | spcegv 3558 | . . . . 5 ⊢ (𝐹 ∈ V → (𝐹:𝑌–onto→𝑋 → ∃𝑧 𝑧:𝑌–onto→𝑋)) |
4 | 3 | imp 408 | . . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → ∃𝑧 𝑧:𝑌–onto→𝑋) |
5 | 4 | olcd 873 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
6 | fof 6760 | . . . . 5 ⊢ (𝐹:𝑌–onto→𝑋 → 𝐹:𝑌⟶𝑋) | |
7 | dmfex 7848 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌⟶𝑋) → 𝑌 ∈ V) | |
8 | 6, 7 | sylan2 594 | . . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → 𝑌 ∈ V) |
9 | brwdom 9511 | . . . 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 257 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ≼* 𝑌) |
12 | 1, 11 | sylan 581 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ≼* 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3447 ∅c0 4286 class class class wbr 5109 ⟶wf 6496 –onto→wfo 6498 ≼* cwdom 9508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-fun 6502 df-fn 6503 df-f 6504 df-fo 6506 df-wdom 9509 |
This theorem is referenced by: wdomref 9516 wdomtr 9519 wdom2d 9524 wdomima2g 9530 ixpiunwdom 9534 harwdom 9535 isf32lem10 10306 fin1a2lem7 10350 |
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