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Theorem fowdom 9640

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 3509	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		foeq1 6830	. . . . . 6 ⊢ (𝑧 = 𝐹 → (𝑧:𝑌–onto→𝑋 ↔ 𝐹:𝑌–onto→𝑋))
3	2	spcegv 3610	. . . . 5 ⊢ (𝐹 ∈ V → (𝐹:𝑌–onto→𝑋 → ∃𝑧 𝑧:𝑌–onto→𝑋))
4	3	imp 406	. . . 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 6834	. . . . 5 ⊢ (𝐹:𝑌–onto→𝑋 → 𝐹:𝑌⟶𝑋)
7		dmfex 7945	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌⟶𝑋) → 𝑌 ∈ V)
8	6, 7	sylan2 592	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → 𝑌 ∈ V)
9		brwdom 9636	. . . 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 579	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ≼^* 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 846 = wceq 1537 ∃wex 1777 ∈ wcel 2108 Vcvv 3488 ∅c0 4352 class class class wbr 5166 ⟶wf 6569 –onto→wfo 6571 ≼^* cwdom 9633

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-fun 6575 df-fn 6576 df-f 6577 df-fo 6579 df-wdom 9634

This theorem is referenced by: wdomref 9641 wdomtr 9644 wdom2d 9649 wdomima2g 9655 ixpiunwdom 9659 harwdom 9660 isf32lem10 10431 fin1a2lem7 10475

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