Theorem indexdom 37713

Description: If for every element of an indexing set 𝐴 there exists a corresponding element of another set 𝐵, then there exists a subset of 𝐵 consisting only of those elements which are indexed by 𝐴, and which is dominated by the set 𝐴. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
indexdom	⊢ ((𝐴 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑐((𝑐 ≼ 𝐴 ∧ 𝑐 ⊆ 𝐵) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑)))

Distinct variable groups:   𝐴,𝑐,𝑥,𝑦   𝐵,𝑐,𝑥,𝑦   𝜑,𝑐

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑀(𝑥,𝑦,𝑐)

Proof of Theorem indexdom

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfsbc1v 3764	. . 3 ⊢ Ⅎ𝑦[(𝑓‘𝑥) / 𝑦]𝜑
2		sbceq1a 3755	. . 3 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ [(𝑓‘𝑥) / 𝑦]𝜑))
3	1, 2	ac6gf 37711	. 2 ⊢ ((𝐴 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))
4		fdm 6665	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴)
5		vex 3442	. . . . . . . 8 ⊢ 𝑓 ∈ V
6	5	dmex 7849	. . . . . . 7 ⊢ dom 𝑓 ∈ V
7	4, 6	eqeltrrdi 2837	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V)
8		ffn 6656	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
9		fnrndomg 10449	. . . . . 6 ⊢ (𝐴 ∈ V → (𝑓 Fn 𝐴 → ran 𝑓 ≼ 𝐴))
10	7, 8, 9	sylc 65	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ≼ 𝐴)
11	10	adantr 480	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ran 𝑓 ≼ 𝐴)
12		frn 6663	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵)
13	12	adantr 480	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ran 𝑓 ⊆ 𝐵)
14		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑥 𝑓:𝐴⟶𝐵
15		nfra1 3253	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑
16	14, 15	nfan 1899	. . . . 5 ⊢ Ⅎ𝑥(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)
17		ffun 6659	. . . . . . . . . 10 ⊢ (𝑓:𝐴⟶𝐵 → Fun 𝑓)
18	17	adantr 480	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → Fun 𝑓)
19	4	eleq2d 2814	. . . . . . . . . 10 ⊢ (𝑓:𝐴⟶𝐵 → (𝑥 ∈ dom 𝑓 ↔ 𝑥 ∈ 𝐴))
20	19	biimpar 477	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝑓)
21		fvelrn 7014	. . . . . . . . 9 ⊢ ((Fun 𝑓 ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ ran 𝑓)
22	18, 20, 21	syl2anc 584	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ran 𝑓)
23	22	adantlr 715	. . . . . . 7 ⊢ (((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ran 𝑓)
24		rspa 3218	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑 ∧ 𝑥 ∈ 𝐴) → [(𝑓‘𝑥) / 𝑦]𝜑)
25	24	adantll 714	. . . . . . 7 ⊢ (((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) ∧ 𝑥 ∈ 𝐴) → [(𝑓‘𝑥) / 𝑦]𝜑)
26		rspesbca 3835	. . . . . . 7 ⊢ (((𝑓‘𝑥) ∈ ran 𝑓 ∧ [(𝑓‘𝑥) / 𝑦]𝜑) → ∃𝑦 ∈ ran 𝑓𝜑)
27	23, 25, 26	syl2anc 584	. . . . . 6 ⊢ (((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ran 𝑓𝜑)
28	27	ex 412	. . . . 5 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ ran 𝑓𝜑))
29	16, 28	ralrimi 3227	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑)
30		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑦 𝑓:𝐴⟶𝐵
31		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
32	31, 1	nfralw 3277	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑
33	30, 32	nfan 1899	. . . . 5 ⊢ Ⅎ𝑦(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)
34		fvelrnb 6887	. . . . . . . 8 ⊢ (𝑓 Fn 𝐴 → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝑦))
35	8, 34	syl 17	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝑦))
36	35	adantr 480	. . . . . 6 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝑦))
37		rsp 3217	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑 → (𝑥 ∈ 𝐴 → [(𝑓‘𝑥) / 𝑦]𝜑))
38	37	adantl 481	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (𝑥 ∈ 𝐴 → [(𝑓‘𝑥) / 𝑦]𝜑))
39	2	eqcoms 2737	. . . . . . . . 9 ⊢ ((𝑓‘𝑥) = 𝑦 → (𝜑 ↔ [(𝑓‘𝑥) / 𝑦]𝜑))
40	39	biimprcd 250	. . . . . . . 8 ⊢ ([(𝑓‘𝑥) / 𝑦]𝜑 → ((𝑓‘𝑥) = 𝑦 → 𝜑))
41	38, 40	syl6 35	. . . . . . 7 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (𝑥 ∈ 𝐴 → ((𝑓‘𝑥) = 𝑦 → 𝜑)))
42	16, 41	reximdai 3231	. . . . . 6 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝑦 → ∃𝑥 ∈ 𝐴 𝜑))
43	36, 42	sylbid 240	. . . . 5 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (𝑦 ∈ ran 𝑓 → ∃𝑥 ∈ 𝐴 𝜑))
44	33, 43	ralrimi 3227	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑)
45	5	rnex 7850	. . . . 5 ⊢ ran 𝑓 ∈ V
46		breq1 5098	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (𝑐 ≼ 𝐴 ↔ ran 𝑓 ≼ 𝐴))
47		sseq1 3963	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (𝑐 ⊆ 𝐵 ↔ ran 𝑓 ⊆ 𝐵))
48	46, 47	anbi12d 632	. . . . . 6 ⊢ (𝑐 = ran 𝑓 → ((𝑐 ≼ 𝐴 ∧ 𝑐 ⊆ 𝐵) ↔ (ran 𝑓 ≼ 𝐴 ∧ ran 𝑓 ⊆ 𝐵)))
49		rexeq 3286	. . . . . . . 8 ⊢ (𝑐 = ran 𝑓 → (∃𝑦 ∈ 𝑐 𝜑 ↔ ∃𝑦 ∈ ran 𝑓𝜑))
50	49	ralbidv 3152	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑))
51		raleq 3287	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑))
52	50, 51	anbi12d 632	. . . . . 6 ⊢ (𝑐 = ran 𝑓 → ((∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑)))
53	48, 52	anbi12d 632	. . . . 5 ⊢ (𝑐 = ran 𝑓 → (((𝑐 ≼ 𝐴 ∧ 𝑐 ⊆ 𝐵) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑)) ↔ ((ran 𝑓 ≼ 𝐴 ∧ ran 𝑓 ⊆ 𝐵) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑))))
54	45, 53	spcev 3563	. . . 4 ⊢ (((ran 𝑓 ≼ 𝐴 ∧ ran 𝑓 ⊆ 𝐵) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑)) → ∃𝑐((𝑐 ≼ 𝐴 ∧ 𝑐 ⊆ 𝐵) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑)))
55	11, 13, 29, 44, 54	syl22anc 838	. . 3 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ∃𝑐((𝑐 ≼ 𝐴 ∧ 𝑐 ⊆ 𝐵) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑)))
56	55	exlimiv 1930	. 2 ⊢ (∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ∃𝑐((𝑐 ≼ 𝐴 ∧ 𝑐 ⊆ 𝐵) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑)))
57	3, 56	syl 17	1 ⊢ ((𝐴 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑐((𝑐 ≼ 𝐴 ∧ 𝑐 ⊆ 𝐵) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3438  [wsbc 3744   ⊆ wss 3905   class class class wbr 5095  dom cdm 5623  ran crn 5624  Fun wfun 6480   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486   ≼ cdom 8877

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-reg 9503  ax-inf2 9556  ax-ac2 10376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-r1 9679  df-rank 9680  df-card 9854  df-acn 9857  df-ac 10029

This theorem is referenced by: (None)