Theorem wdomima2g 9073

Description: A set is weakly dominant over its image under any function. This version of wdomimag 9074 is stated so as to avoid ax-rep 5154. (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 5535	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		funres 6375	. . . . . . . 8 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
3		funforn 6581	. . . . . . . 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 1131	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴))
6		fof 6574	. . . . . 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 5843	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
9		inss1 4134	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
10	8, 9	eqsstri 3927	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
11		simp2 1135	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → 𝐴 ∈ 𝑉)
12		ssexg 5191	. . . . . 6 ⊢ ((dom (𝐹 ↾ 𝐴) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom (𝐹 ↾ 𝐴) ∈ V)
13	10, 11, 12	sylancr 591	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → dom (𝐹 ↾ 𝐴) ∈ V)
14		simp3 1136	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 “ 𝐴) ∈ 𝑊)
15	1, 14	eqeltrrid 2858	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → ran (𝐹 ↾ 𝐴) ∈ 𝑊)
16		fex2 7641	. . . . 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 1369	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 ↾ 𝐴) ∈ V)
18		fowdom 9058	. . . 4 ⊢ (((𝐹 ↾ 𝐴) ∈ V ∧ (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴)) → ran (𝐹 ↾ 𝐴) ≼* dom (𝐹 ↾ 𝐴))
19	17, 5, 18	syl2anc 588	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → ran (𝐹 ↾ 𝐴) ≼* dom (𝐹 ↾ 𝐴))
20		ssdomg 8571	. . . . . 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 9061	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) ≼ 𝐴 → dom (𝐹 ↾ 𝐴) ≼* 𝐴)
23	21, 22	syl 17	. . . 4 ⊢ (𝐴 ∈ 𝑉 → dom (𝐹 ↾ 𝐴) ≼* 𝐴)
24	23	3ad2ant2 1132	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → dom (𝐹 ↾ 𝐴) ≼* 𝐴)
25		wdomtr 9062	. . 3 ⊢ ((ran (𝐹 ↾ 𝐴) ≼* dom (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) ≼* 𝐴) → ran (𝐹 ↾ 𝐴) ≼* 𝐴)
26	19, 24, 25	syl2anc 588	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → ran (𝐹 ↾ 𝐴) ≼* 𝐴)
27	1, 26	eqbrtrid 5065	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 “ 𝐴) ≼* 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1085   ∈ wcel 2112  Vcvv 3410   ∩ cin 3858   ⊆ wss 3859   class class class wbr 5030  dom cdm 5522  ran crn 5523   ↾ cres 5524   “ cima 5525  Fun wfun 6327  ⟶wf 6329  –onto→wfo 6331   ≼ cdom 8523   ≼^* cwdom 9051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-op 4527  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-er 8297  df-en 8526  df-dom 8527  df-sdom 8528  df-wdom 9052

This theorem is referenced by:  wdomimag  9074  unxpwdom2  9075