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Theorem fowdom 8718

Description: An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)

**Assertion**
Ref	Expression
fowdom	⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ≼^* 𝑌)

Proof of Theorem fowdom Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3400	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		foeq1 6327	. . . . . 6 ⊢ (𝑧 = 𝐹 → (𝑧:𝑌–onto→𝑋 ↔ 𝐹:𝑌–onto→𝑋))
3	2	spcegv 3482	. . . . 5 ⊢ (𝐹 ∈ V → (𝐹:𝑌–onto→𝑋 → ∃𝑧 𝑧:𝑌–onto→𝑋))
4	3	imp 396	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → ∃𝑧 𝑧:𝑌–onto→𝑋)
5	4	olcd 901	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))
6		fof 6331	. . . . 5 ⊢ (𝐹:𝑌–onto→𝑋 → 𝐹:𝑌⟶𝑋)
7		dmfex 7359	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌⟶𝑋) → 𝑌 ∈ V)
8	6, 7	sylan2 587	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → 𝑌 ∈ V)
9		brwdom 8714	. . . 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 249	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ≼^* 𝑌)
12	1, 11	sylan 576	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ≼^* 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∨ wo 874 = wceq 1653 ∃wex 1875 ∈ wcel 2157 Vcvv 3385 ∅c0 4115 class class class wbr 4843 ⟶wf 6097 –onto→wfo 6099 ≼^* cwdom 8704

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 ax-un 7183

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-fun 6103 df-fn 6104 df-f 6105 df-fo 6107 df-wdom 8706

This theorem is referenced by: wdomref 8719 wdomtr 8722 wdom2d 8727 wdomima2g 8733 harwdom 8737 ixpiunwdom 8738 isf32lem10 9472 fin1a2lem7 9516

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