Theorem wdomima2g 9626

Description: A set is weakly dominant over its image under any function. This version of wdomimag 9627 is stated so as to avoid ax-rep 5279. (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 5698	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		funres 6608	. . . . . . . 8 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
3		funforn 6827	. . . . . . . 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 6820	. . . . . 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 6030	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
9		inss1 4237	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
10	8, 9	eqsstri 4030	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
11		simp2 1138	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → 𝐴 ∈ 𝑉)
12		ssexg 5323	. . . . . 6 ⊢ ((dom (𝐹 ↾ 𝐴) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom (𝐹 ↾ 𝐴) ∈ V)
13	10, 11, 12	sylancr 587	. . . . 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 2846	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → ran (𝐹 ↾ 𝐴) ∈ 𝑊)
16		fex2 7958	. . . . 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 1373	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 ↾ 𝐴) ∈ V)
18		fowdom 9611	. . . 4 ⊢ (((𝐹 ↾ 𝐴) ∈ V ∧ (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴)) → ran (𝐹 ↾ 𝐴) ≼* dom (𝐹 ↾ 𝐴))
19	17, 5, 18	syl2anc 584	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → ran (𝐹 ↾ 𝐴) ≼* dom (𝐹 ↾ 𝐴))
20		ssdomg 9040	. . . . . 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 9614	. . . . 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 9615	. . 3 ⊢ ((ran (𝐹 ↾ 𝐴) ≼* dom (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) ≼* 𝐴) → ran (𝐹 ↾ 𝐴) ≼* 𝐴)
26	19, 24, 25	syl2anc 584	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → ran (𝐹 ↾ 𝐴) ≼* 𝐴)
27	1, 26	eqbrtrid 5178	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 “ 𝐴) ≼* 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2108  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951   class class class wbr 5143  dom cdm 5685  ran crn 5686   ↾ cres 5687   “ cima 5688  Fun wfun 6555  ⟶wf 6557  –onto→wfo 6559   ≼ cdom 8983   ≼^* cwdom 9604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-en 8986  df-dom 8987  df-sdom 8988  df-wdom 9605

This theorem is referenced by:  wdomimag  9627  unxpwdom2  9628