Theorem wdomima2g 9544

Description: A set is weakly dominant over its image under any function. This version of wdomimag 9545 is stated so as to avoid ax-rep 5238. (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 5674	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		funres 6578	. . . . . . . 8 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
3		funforn 6799	. . . . . . . 8 ⊢ (Fun (𝐹 ↾ 𝐴) ↔ (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴))
4	2, 3	sylib 221	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴))
5	4	3ad2ant1 1151	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴))
6		fof 6792	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴) → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶ran (𝐹 ↾ 𝐴))
7	5, 6	syl 18	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)⟶ran (𝐹 ↾ 𝐴))
8		dmres 6011	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
9		inss1 4189	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
10	8, 9	eqsstri 3983	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
11		simp2 1155	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → 𝐴 ∈ 𝑉)
12		ssexg 5290	. . . . . 6 ⊢ ((dom (𝐹 ↾ 𝐴) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom (𝐹 ↾ 𝐴) ∈ V)
13	10, 11, 12	sylancr 598	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → dom (𝐹 ↾ 𝐴) ∈ V)
14		simp3 1156	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 “ 𝐴) ∈ 𝑊)
15	1, 14	eqeltrrid 2868	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → ran (𝐹 ↾ 𝐴) ∈ 𝑊)
16		fex2 7929	. . . . 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 1398	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 ↾ 𝐴) ∈ V)
18		fowdom 9529	. . . 4 ⊢ (((𝐹 ↾ 𝐴) ∈ V ∧ (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴)) → ran (𝐹 ↾ 𝐴) ≼* dom (𝐹 ↾ 𝐴))
19	17, 5, 18	syl2anc 595	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → ran (𝐹 ↾ 𝐴) ≼* dom (𝐹 ↾ 𝐴))
20		ssdomg 8993	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (dom (𝐹 ↾ 𝐴) ⊆ 𝐴 → dom (𝐹 ↾ 𝐴) ≼ 𝐴))
21	10, 20	mpi 21	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → dom (𝐹 ↾ 𝐴) ≼ 𝐴)
22		domwdom 9532	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) ≼ 𝐴 → dom (𝐹 ↾ 𝐴) ≼* 𝐴)
23	21, 22	syl 18	. . . 4 ⊢ (𝐴 ∈ 𝑉 → dom (𝐹 ↾ 𝐴) ≼* 𝐴)
24	23	3ad2ant2 1152	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → dom (𝐹 ↾ 𝐴) ≼* 𝐴)
25		wdomtr 9533	. . 3 ⊢ ((ran (𝐹 ↾ 𝐴) ≼* dom (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) ≼* 𝐴) → ran (𝐹 ↾ 𝐴) ≼* 𝐴)
26	19, 24, 25	syl2anc 595	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → ran (𝐹 ↾ 𝐴) ≼* 𝐴)
27	1, 26	eqbrtrid 5146	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 “ 𝐴) ≼* 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1103   ∈ wcel 2143  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905   class class class wbr 5109  dom cdm 5661  ran crn 5662   ↾ cres 5663   “ cima 5664  Fun wfun 6530  ⟶wf 6532  –onto→wfo 6534   ≼ cdom 8937   ≼^* cwdom 9522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-en 8940  df-dom 8941  df-sdom 8942  df-wdom 9523

This theorem is referenced by:  wdomimag  9545  unxpwdom2  9546