Theorem indexdom 37906

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 3761	. . 3 ⊢ Ⅎ𝑦[(𝑓‘𝑥) / 𝑦]𝜑
2		sbceq1a 3752	. . 3 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ [(𝑓‘𝑥) / 𝑦]𝜑))
3	1, 2	ac6gf 37904	. 2 ⊢ ((𝐴 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))
4		fdm 6672	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴)
5		vex 3445	. . . . . . . 8 ⊢ 𝑓 ∈ V
6	5	dmex 7853	. . . . . . 7 ⊢ dom 𝑓 ∈ V
7	4, 6	eqeltrrdi 2846	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V)
8		ffn 6663	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
9		fnrndomg 10450	. . . . . 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 6670	. . . . 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 1916	. . . . . 6 ⊢ Ⅎ𝑥 𝑓:𝐴⟶𝐵
15		nfra1 3261	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑
16	14, 15	nfan 1901	. . . . 5 ⊢ Ⅎ𝑥(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)
17		ffun 6666	. . . . . . . . . 10 ⊢ (𝑓:𝐴⟶𝐵 → Fun 𝑓)
18	17	adantr 480	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → Fun 𝑓)
19	4	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑓:𝐴⟶𝐵 → (𝑥 ∈ dom 𝑓 ↔ 𝑥 ∈ 𝐴))
20	19	biimpar 477	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝑓)
21		fvelrn 7023	. . . . . . . . 9 ⊢ ((Fun 𝑓 ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ ran 𝑓)
22	18, 20, 21	syl2anc 585	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ran 𝑓)
23	22	adantlr 716	. . . . . . 7 ⊢ (((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ran 𝑓)
24		rspa 3226	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑 ∧ 𝑥 ∈ 𝐴) → [(𝑓‘𝑥) / 𝑦]𝜑)
25	24	adantll 715	. . . . . . 7 ⊢ (((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) ∧ 𝑥 ∈ 𝐴) → [(𝑓‘𝑥) / 𝑦]𝜑)
26		rspesbca 3832	. . . . . . 7 ⊢ (((𝑓‘𝑥) ∈ ran 𝑓 ∧ [(𝑓‘𝑥) / 𝑦]𝜑) → ∃𝑦 ∈ ran 𝑓𝜑)
27	23, 25, 26	syl2anc 585	. . . . . 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 3235	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑)
30		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑦 𝑓:𝐴⟶𝐵
31		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
32	31, 1	nfralw 3284	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑
33	30, 32	nfan 1901	. . . . 5 ⊢ Ⅎ𝑦(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)
34		fvelrnb 6895	. . . . . . . 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 3225	. . . . . . . . 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 2745	. . . . . . . . 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 3239	. . . . . 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 3235	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑)
45	5	rnex 7854	. . . . 5 ⊢ ran 𝑓 ∈ V
46		breq1 5102	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (𝑐 ≼ 𝐴 ↔ ran 𝑓 ≼ 𝐴))
47		sseq1 3960	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (𝑐 ⊆ 𝐵 ↔ ran 𝑓 ⊆ 𝐵))
48	46, 47	anbi12d 633	. . . . . 6 ⊢ (𝑐 = ran 𝑓 → ((𝑐 ≼ 𝐴 ∧ 𝑐 ⊆ 𝐵) ↔ (ran 𝑓 ≼ 𝐴 ∧ ran 𝑓 ⊆ 𝐵)))
49		rexeq 3293	. . . . . . . 8 ⊢ (𝑐 = ran 𝑓 → (∃𝑦 ∈ 𝑐 𝜑 ↔ ∃𝑦 ∈ ran 𝑓𝜑))
50	49	ralbidv 3160	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑))
51		raleq 3294	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑))
52	50, 51	anbi12d 633	. . . . . 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 633	. . . . 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 3561	. . . 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 839	. . 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 1932	. 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 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061  Vcvv 3441  [wsbc 3741   ⊆ wss 3902   class class class wbr 5099  dom cdm 5625  ran crn 5626  Fun wfun 6487   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493   ≼ cdom 8885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-reg 9501  ax-inf2 9554  ax-ac2 10377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-r1 9680  df-rank 9681  df-card 9855  df-acn 9858  df-ac 10030

This theorem is referenced by: (None)