Theorem wdomima2g 9501

Description: A set is weakly dominant over its image under any function. This version of wdomimag 9502 is stated so as to avoid ax-rep 5212. (Contributed by Mario Carneiro, 25-Jun-2015.)

Assertion

Ref	Expression
wdomima2g	⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 “ 𝐴) ≼* 𝐴)

Proof of Theorem wdomima2g

Step	Hyp	Ref	Expression
1		df-ima 5644	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		funres 6540	. . . . . . . 8 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
3		funforn 6759	. . . . . . . 8 ⊢ (Fun (𝐹 ↾ 𝐴) ↔ (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴))
4	2, 3	sylib 218	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴))
5	4	3ad2ant1 1134	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴))
6		fof 6752	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴) → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶ran (𝐹 ↾ 𝐴))
7	5, 6	syl 17	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶ran (𝐹 ↾ 𝐴))
8		dmres 5977	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
9		inss1 4177	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
10	8, 9	eqsstri 3968	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
11		simp2 1138	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → 𝐴 ∈ 𝑉)
12		ssexg 5264	. . . . . 6 ⊢ ((dom (𝐹 ↾ 𝐴) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom (𝐹 ↾ 𝐴) ∈ V)
13	10, 11, 12	sylancr 588	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → dom (𝐹 ↾ 𝐴) ∈ V)
14		simp3 1139	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 “ 𝐴) ∈ 𝑊)
15	1, 14	eqeltrrid 2841	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → ran (𝐹 ↾ 𝐴) ∈ 𝑊)
16		fex2 7887	. . . . 5 ⊢ (((𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶ran (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) ∈ V ∧ ran (𝐹 ↾ 𝐴) ∈ 𝑊) → (𝐹 ↾ 𝐴) ∈ V)
17	7, 13, 15, 16	syl3anc 1374	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 ↾ 𝐴) ∈ V)
18		fowdom 9486	. . . 4 ⊢ (((𝐹 ↾ 𝐴) ∈ V ∧ (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴)) → ran (𝐹 ↾ 𝐴) ≼* dom (𝐹 ↾ 𝐴))
19	17, 5, 18	syl2anc 585	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → ran (𝐹 ↾ 𝐴) ≼* dom (𝐹 ↾ 𝐴))
20		ssdomg 8947	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (dom (𝐹 ↾ 𝐴) ⊆ 𝐴 → dom (𝐹 ↾ 𝐴) ≼ 𝐴))
21	10, 20	mpi 20	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → dom (𝐹 ↾ 𝐴) ≼ 𝐴)
22		domwdom 9489	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) ≼ 𝐴 → dom (𝐹 ↾ 𝐴) ≼* 𝐴)
23	21, 22	syl 17	. . . 4 ⊢ (𝐴 ∈ 𝑉 → dom (𝐹 ↾ 𝐴) ≼* 𝐴)
24	23	3ad2ant2 1135	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → dom (𝐹 ↾ 𝐴) ≼* 𝐴)
25		wdomtr 9490	. . 3 ⊢ ((ran (𝐹 ↾ 𝐴) ≼* dom (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) ≼* 𝐴) → ran (𝐹 ↾ 𝐴) ≼* 𝐴)
26	19, 24, 25	syl2anc 585	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → ran (𝐹 ↾ 𝐴) ≼* 𝐴)
27	1, 26	eqbrtrid 5120	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 “ 𝐴) ≼* 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2114  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889   class class class wbr 5085  dom cdm 5631  ran crn 5632   ↾ cres 5633   “ cima 5634  Fun wfun 6492  ⟶wf 6494  –onto→wfo 6496   ≼ cdom 8891   ≼^* cwdom 9479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-en 8894  df-dom 8895  df-sdom 8896  df-wdom 9480

This theorem is referenced by:  wdomimag  9502  unxpwdom2  9503