Theorem wdomima2g 9655

Description: A set is weakly dominant over its image under any function. This version of wdomimag 9656 is stated so as to avoid ax-rep 5303. (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 5713	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		funres 6620	. . . . . . . 8 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
3		funforn 6841	. . . . . . . 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 1133	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴))
6		fof 6834	. . . . . 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 6041	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
9		inss1 4258	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
10	8, 9	eqsstri 4043	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
11		simp2 1137	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → 𝐴 ∈ 𝑉)
12		ssexg 5341	. . . . . 6 ⊢ ((dom (𝐹 ↾ 𝐴) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom (𝐹 ↾ 𝐴) ∈ V)
13	10, 11, 12	sylancr 586	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → dom (𝐹 ↾ 𝐴) ∈ V)
14		simp3 1138	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 “ 𝐴) ∈ 𝑊)
15	1, 14	eqeltrrid 2849	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → ran (𝐹 ↾ 𝐴) ∈ 𝑊)
16		fex2 7974	. . . . 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 1371	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 ↾ 𝐴) ∈ V)
18		fowdom 9640	. . . 4 ⊢ (((𝐹 ↾ 𝐴) ∈ V ∧ (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴)) → ran (𝐹 ↾ 𝐴) ≼* dom (𝐹 ↾ 𝐴))
19	17, 5, 18	syl2anc 583	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → ran (𝐹 ↾ 𝐴) ≼* dom (𝐹 ↾ 𝐴))
20		ssdomg 9060	. . . . . 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 9643	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) ≼ 𝐴 → dom (𝐹 ↾ 𝐴) ≼* 𝐴)
23	21, 22	syl 17	. . . 4 ⊢ (𝐴 ∈ 𝑉 → dom (𝐹 ↾ 𝐴) ≼* 𝐴)
24	23	3ad2ant2 1134	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → dom (𝐹 ↾ 𝐴) ≼* 𝐴)
25		wdomtr 9644	. . 3 ⊢ ((ran (𝐹 ↾ 𝐴) ≼* dom (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) ≼* 𝐴) → ran (𝐹 ↾ 𝐴) ≼* 𝐴)
26	19, 24, 25	syl2anc 583	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → ran (𝐹 ↾ 𝐴) ≼* 𝐴)
27	1, 26	eqbrtrid 5201	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 “ 𝐴) ≼* 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2108  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976   class class class wbr 5166  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703  Fun wfun 6567  ⟶wf 6569  –onto→wfo 6571   ≼ cdom 9001   ≼^* 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-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  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-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  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-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-en 9004  df-dom 9005  df-sdom 9006  df-wdom 9634

This theorem is referenced by:  wdomimag  9656  unxpwdom2  9657