Theorem indexdom 37358

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 3793	. . 3 ⊢ Ⅎ𝑦[(𝑓‘𝑥) / 𝑦]𝜑
2		sbceq1a 3784	. . 3 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ [(𝑓‘𝑥) / 𝑦]𝜑))
3	1, 2	ac6gf 37356	. 2 ⊢ ((𝐴 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))
4		fdm 6732	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴)
5		vex 3465	. . . . . . . 8 ⊢ 𝑓 ∈ V
6	5	dmex 7917	. . . . . . 7 ⊢ dom 𝑓 ∈ V
7	4, 6	eqeltrrdi 2834	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V)
8		ffn 6723	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
9		fnrndomg 10566	. . . . . 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 479	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ran 𝑓 ≼ 𝐴)
12		frn 6730	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵)
13	12	adantr 479	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ran 𝑓 ⊆ 𝐵)
14		nfv 1909	. . . . . 6 ⊢ Ⅎ𝑥 𝑓:𝐴⟶𝐵
15		nfra1 3271	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑
16	14, 15	nfan 1894	. . . . 5 ⊢ Ⅎ𝑥(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)
17		ffun 6726	. . . . . . . . . 10 ⊢ (𝑓:𝐴⟶𝐵 → Fun 𝑓)
18	17	adantr 479	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → Fun 𝑓)
19	4	eleq2d 2811	. . . . . . . . . 10 ⊢ (𝑓:𝐴⟶𝐵 → (𝑥 ∈ dom 𝑓 ↔ 𝑥 ∈ 𝐴))
20	19	biimpar 476	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝑓)
21		fvelrn 7085	. . . . . . . . 9 ⊢ ((Fun 𝑓 ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ ran 𝑓)
22	18, 20, 21	syl2anc 582	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ran 𝑓)
23	22	adantlr 713	. . . . . . 7 ⊢ (((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ran 𝑓)
24		rspa 3235	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑 ∧ 𝑥 ∈ 𝐴) → [(𝑓‘𝑥) / 𝑦]𝜑)
25	24	adantll 712	. . . . . . 7 ⊢ (((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) ∧ 𝑥 ∈ 𝐴) → [(𝑓‘𝑥) / 𝑦]𝜑)
26		rspesbca 3871	. . . . . . 7 ⊢ (((𝑓‘𝑥) ∈ ran 𝑓 ∧ [(𝑓‘𝑥) / 𝑦]𝜑) → ∃𝑦 ∈ ran 𝑓𝜑)
27	23, 25, 26	syl2anc 582	. . . . . 6 ⊢ (((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ran 𝑓𝜑)
28	27	ex 411	. . . . 5 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ ran 𝑓𝜑))
29	16, 28	ralrimi 3244	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑)
30		nfv 1909	. . . . . 6 ⊢ Ⅎ𝑦 𝑓:𝐴⟶𝐵
31		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
32	31, 1	nfralw 3298	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑
33	30, 32	nfan 1894	. . . . 5 ⊢ Ⅎ𝑦(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)
34		fvelrnb 6958	. . . . . . . 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 479	. . . . . 6 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝑦))
37		rsp 3234	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑 → (𝑥 ∈ 𝐴 → [(𝑓‘𝑥) / 𝑦]𝜑))
38	37	adantl 480	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (𝑥 ∈ 𝐴 → [(𝑓‘𝑥) / 𝑦]𝜑))
39	2	eqcoms 2733	. . . . . . . . 9 ⊢ ((𝑓‘𝑥) = 𝑦 → (𝜑 ↔ [(𝑓‘𝑥) / 𝑦]𝜑))
40	39	biimprcd 249	. . . . . . . 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 3248	. . . . . 6 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝑦 → ∃𝑥 ∈ 𝐴 𝜑))
43	36, 42	sylbid 239	. . . . 5 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (𝑦 ∈ ran 𝑓 → ∃𝑥 ∈ 𝐴 𝜑))
44	33, 43	ralrimi 3244	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑)
45	5	rnex 7918	. . . . 5 ⊢ ran 𝑓 ∈ V
46		breq1 5152	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (𝑐 ≼ 𝐴 ↔ ran 𝑓 ≼ 𝐴))
47		sseq1 4002	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (𝑐 ⊆ 𝐵 ↔ ran 𝑓 ⊆ 𝐵))
48	46, 47	anbi12d 630	. . . . . 6 ⊢ (𝑐 = ran 𝑓 → ((𝑐 ≼ 𝐴 ∧ 𝑐 ⊆ 𝐵) ↔ (ran 𝑓 ≼ 𝐴 ∧ ran 𝑓 ⊆ 𝐵)))
49		rexeq 3310	. . . . . . . 8 ⊢ (𝑐 = ran 𝑓 → (∃𝑦 ∈ 𝑐 𝜑 ↔ ∃𝑦 ∈ ran 𝑓𝜑))
50	49	ralbidv 3167	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑))
51		raleq 3311	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑))
52	50, 51	anbi12d 630	. . . . . 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 630	. . . . 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 3590	. . . 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 837	. . 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 1925	. 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 205   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∀wral 3050  ∃wrex 3059  Vcvv 3461  [wsbc 3773   ⊆ wss 3944   class class class wbr 5149  dom cdm 5678  ran crn 5679  Fun wfun 6543   Fn wfn 6544  ⟶wf 6545  ‘cfv 6549   ≼ cdom 8962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-reg 9622  ax-inf2 9671  ax-ac2 10493

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-r1 9794  df-rank 9795  df-card 9969  df-acn 9972  df-ac 10146

This theorem is referenced by: (None)