Theorem indexdom 38351

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 3763	. . 3 ⊢ Ⅎ𝑦[(𝑓‘𝑥) / 𝑦]𝜑
2		sbceq1a 3754	. . 3 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ [(𝑓‘𝑥) / 𝑦]𝜑))
3	1, 2	ac6gf 38349	. 2 ⊢ ((𝐴 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))
4		fdm 6715	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴)
5		vex 3457	. . . . . . . 8 ⊢ 𝑓 ∈ V
6	5	dmex 7905	. . . . . . 7 ⊢ dom 𝑓 ∈ V
7	4, 6	eqeltrrdi 2870	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V)
8		ffn 6705	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
9		fnrndomg 10519	. . . . . 6 ⊢ (𝐴 ∈ V → (𝑓 Fn 𝐴 → ran 𝑓 ≼ 𝐴))
10	7, 8, 9	sylc 66	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ≼ 𝐴)
11	10	adantr 485	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ran 𝑓 ≼ 𝐴)
12		frn 6713	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵)
13	12	adantr 485	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ran 𝑓 ⊆ 𝐵)
14		nfv 1942	. . . . . 6 ⊢ Ⅎ𝑥 𝑓:𝐴⟶𝐵
15		nfra1 3287	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑
16	14, 15	nfan 1927	. . . . 5 ⊢ Ⅎ𝑥(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)
17		ffun 6708	. . . . . . . . . 10 ⊢ (𝑓:𝐴⟶𝐵 → Fun 𝑓)
18	17	adantr 485	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → Fun 𝑓)
19	4	eleq2d 2847	. . . . . . . . . 10 ⊢ (𝑓:𝐴⟶𝐵 → (𝑥 ∈ dom 𝑓 ↔ 𝑥 ∈ 𝐴))
20	19	biimpar 482	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝑓)
21		fvelrn 7071	. . . . . . . . 9 ⊢ ((Fun 𝑓 ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ ran 𝑓)
22	18, 20, 21	syl2anc 595	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ran 𝑓)
23	22	adantlr 727	. . . . . . 7 ⊢ (((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ran 𝑓)
24		rspa 3252	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑 ∧ 𝑥 ∈ 𝐴) → [(𝑓‘𝑥) / 𝑦]𝜑)
25	24	adantll 726	. . . . . . 7 ⊢ (((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) ∧ 𝑥 ∈ 𝐴) → [(𝑓‘𝑥) / 𝑦]𝜑)
26		rspesbca 3833	. . . . . . 7 ⊢ (((𝑓‘𝑥) ∈ ran 𝑓 ∧ [(𝑓‘𝑥) / 𝑦]𝜑) → ∃𝑦 ∈ ran 𝑓𝜑)
27	23, 25, 26	syl2anc 595	. . . . . 6 ⊢ (((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ran 𝑓𝜑)
28	27	ex 417	. . . . 5 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ ran 𝑓𝜑))
29	16, 28	ralrimi 3261	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑)
30		nfv 1942	. . . . . 6 ⊢ Ⅎ𝑦 𝑓:𝐴⟶𝐵
31		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
32	31, 1	nfralw 3310	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑
33	30, 32	nfan 1927	. . . . 5 ⊢ Ⅎ𝑦(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)
34		fvelrnb 6941	. . . . . . . 8 ⊢ (𝑓 Fn 𝐴 → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝑦))
35	8, 34	syl 18	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝑦))
36	35	adantr 485	. . . . . 6 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝑦))
37		rsp 3251	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑 → (𝑥 ∈ 𝐴 → [(𝑓‘𝑥) / 𝑦]𝜑))
38	37	adantl 486	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (𝑥 ∈ 𝐴 → [(𝑓‘𝑥) / 𝑦]𝜑))
39	2	eqcoms 2769	. . . . . . . . 9 ⊢ ((𝑓‘𝑥) = 𝑦 → (𝜑 ↔ [(𝑓‘𝑥) / 𝑦]𝜑))
40	39	biimprcd 253	. . . . . . . 8 ⊢ ([(𝑓‘𝑥) / 𝑦]𝜑 → ((𝑓‘𝑥) = 𝑦 → 𝜑))
41	38, 40	syl6 36	. . . . . . 7 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (𝑥 ∈ 𝐴 → ((𝑓‘𝑥) = 𝑦 → 𝜑)))
42	16, 41	reximdai 3265	. . . . . 6 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝑦 → ∃𝑥 ∈ 𝐴 𝜑))
43	36, 42	sylbid 243	. . . . 5 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → (𝑦 ∈ ran 𝑓 → ∃𝑥 ∈ 𝐴 𝜑))
44	33, 43	ralrimi 3261	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑)
45	5	rnex 7906	. . . . 5 ⊢ ran 𝑓 ∈ V
46		breq1 5111	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (𝑐 ≼ 𝐴 ↔ ran 𝑓 ≼ 𝐴))
47		sseq1 3961	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (𝑐 ⊆ 𝐵 ↔ ran 𝑓 ⊆ 𝐵))
48	46, 47	anbi12d 643	. . . . . 6 ⊢ (𝑐 = ran 𝑓 → ((𝑐 ≼ 𝐴 ∧ 𝑐 ⊆ 𝐵) ↔ (ran 𝑓 ≼ 𝐴 ∧ ran 𝑓 ⊆ 𝐵)))
49		rexeq 3317	. . . . . . . 8 ⊢ (𝑐 = ran 𝑓 → (∃𝑦 ∈ 𝑐 𝜑 ↔ ∃𝑦 ∈ ran 𝑓𝜑))
50	49	ralbidv 3186	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑))
51		raleq 3318	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑))
52	50, 51	anbi12d 643	. . . . . 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 643	. . . . 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 3564	. . . 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 851	. . 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 1958	. 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 18	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 209   ∧ wa 400   = wceq 1568  ∃wex 1807   ∈ wcel 2141  ∀wral 3077  ∃wrex 3087  Vcvv 3453  [wsbc 3743   ⊆ wss 3904   class class class wbr 5108  dom cdm 5661  ran crn 5662  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536   ≼ cdom 8940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-reg 9553  ax-inf2 9609  ax-ac2 10446

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  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-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-r1 9735  df-rank 9736  df-card 9924  df-acn 9927  df-ac 10099

This theorem is referenced by: (None)