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Theorem fowdom 9602

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

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

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

Step	Hyp	Ref	Expression
1		elex 3475	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		foeq1 6822	. . . . . 6 ⊢ (𝑧 = 𝐹 → (𝑧:𝑌⟶onto𝑋 ↔ 𝐹:𝑌⟶onto𝑋))
3	2	spcegv 3555	. . . . 5 ⊢ (𝐹 ∈ V → (𝐹:𝑌⟶onto𝑋 → ∃𝑧 𝑧:𝑌⟶onto𝑋))
4	3	imp 412	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌⟶onto𝑋) → ∃𝑧 𝑧:𝑌⟶onto𝑋)
5	4	olcd 888	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌⟶onto𝑋) → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌⟶onto𝑋))
6		fof 6826	. . . . 5 ⊢ (𝐹:𝑌⟶onto𝑋 → 𝐹:𝑌⟶𝑋)
7		dmfex 7951	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌⟶𝑋) → 𝑌 ∈ V)
8	6, 7	sylan2 605	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌⟶onto𝑋) → 𝑌 ∈ V)
9		brwdom 9598	. . . 4 ⊢ (𝑌 ∈ V → (𝑋 ≼^* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌⟶onto𝑋)))
10	8, 9	syl 18	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌⟶onto𝑋) → (𝑋 ≼^* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌⟶onto𝑋)))
11	5, 10	mpbird 260	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌⟶onto𝑋) → 𝑋 ≼^* 𝑌)
12	1, 11	sylan 592	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝑌⟶onto𝑋) → 𝑋 ≼^* 𝑌)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∃wex 1812 ∈ wcel 2147 Vcvv 3454 ∅c0 4281 class class class wbr 5108 ⟶wf 6562 ⟶ontowfo 6564 ≼^* cwdom 9595

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pr 5398 ax-un 7783

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2102 df-clab 2742 df-cleq 2755 df-clel 2839 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-br 5109 df-opab 5174 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-fun 6568 df-fn 6569 df-f 6570 df-fo 6572 df-wdom 9596

This theorem is used by: wdomref 9603 wdomtr 9606 wdom2d 9611 wdomima2g 9617 ixpiunwdom 9621 harwdom 9622 isf32lem10 10479 fin1a2lem7 10523

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