Theorem fodomfi2 9480

Description: Onto functions define dominance when a finite number of choices need to be made. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Assertion

Ref	Expression
fodomfi2	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴)

Proof of Theorem fodomfi2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fofn 6587	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
2	1	3ad2ant3 1131	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐹 Fn 𝐴)
3		forn 6588	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
4		eqimss2 4024	. . . . 5 ⊢ (ran 𝐹 = 𝐵 → 𝐵 ⊆ ran 𝐹)
5	3, 4	syl 17	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐵 ⊆ ran 𝐹)
6	5	3ad2ant3 1131	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ⊆ ran 𝐹)
7		simp2 1133	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ Fin)
8		fipreima 8824	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ∧ 𝐵 ∈ Fin) → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)(𝐹 “ 𝑥) = 𝐵)
9	2, 6, 7, 8	syl3anc 1367	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)(𝐹 “ 𝑥) = 𝐵)
10		elinel2 4173	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ Fin)
11	10	adantl 484	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ∈ Fin)
12		finnum 9371	. . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card)
13	11, 12	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ∈ dom card)
14		simpl3 1189	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹:𝐴–onto→𝐵)
15		fofun 6586	. . . . . . . 8 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹)
16	14, 15	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → Fun 𝐹)
17		elinel1 4172	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴)
18	17	elpwid 4553	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴)
19	18	adantl 484	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ⊆ 𝐴)
20		fof 6585	. . . . . . . . 9 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
21		fdm 6517	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
22	14, 20, 21	3syl 18	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → dom 𝐹 = 𝐴)
23	19, 22	sseqtrrd 4008	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ⊆ dom 𝐹)
24		fores 6595	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ⊆ dom 𝐹) → (𝐹 ↾ 𝑥):𝑥–onto→(𝐹 “ 𝑥))
25	16, 23, 24	syl2anc 586	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ 𝑥):𝑥–onto→(𝐹 “ 𝑥))
26		fodomnum 9477	. . . . . 6 ⊢ (𝑥 ∈ dom card → ((𝐹 ↾ 𝑥):𝑥–onto→(𝐹 “ 𝑥) → (𝐹 “ 𝑥) ≼ 𝑥))
27	13, 25, 26	sylc 65	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ 𝑥) ≼ 𝑥)
28		simpl1 1187	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐴 ∈ 𝑉)
29		ssdomg 8549	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ⊆ 𝐴 → 𝑥 ≼ 𝐴))
30	28, 19, 29	sylc 65	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ≼ 𝐴)
31		domtr 8556	. . . . 5 ⊢ (((𝐹 “ 𝑥) ≼ 𝑥 ∧ 𝑥 ≼ 𝐴) → (𝐹 “ 𝑥) ≼ 𝐴)
32	27, 30, 31	syl2anc 586	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ 𝑥) ≼ 𝐴)
33		breq1 5062	. . . 4 ⊢ ((𝐹 “ 𝑥) = 𝐵 → ((𝐹 “ 𝑥) ≼ 𝐴 ↔ 𝐵 ≼ 𝐴))
34	32, 33	syl5ibcom 247	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐹 “ 𝑥) = 𝐵 → 𝐵 ≼ 𝐴))
35	34	rexlimdva 3284	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)(𝐹 “ 𝑥) = 𝐵 → 𝐵 ≼ 𝐴))
36	9, 35	mpd 15	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∃wrex 3139   ∩ cin 3935   ⊆ wss 3936  𝒫 cpw 4539   class class class wbr 5059  dom cdm 5550  ran crn 5551   ↾ cres 5552   “ cima 5553  Fun wfun 6344   Fn wfn 6345  ⟶wf 6346  –onto→wfo 6348   ≼ cdom 8501  Fincfn 8503  cardccrd 9358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-1o 8096  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-fin 8507  df-card 9362  df-acn 9365

This theorem is referenced by:  wdomfil  9481