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Theorem indexdom 35130
 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 ((𝐴𝑀 ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑐((𝑐𝐴𝑐𝐵) ∧ (∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 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.
StepHypRef Expression
1 nfsbc1v 3767 . . 3 𝑦[(𝑓𝑥) / 𝑦]𝜑
2 sbceq1a 3758 . . 3 (𝑦 = (𝑓𝑥) → (𝜑[(𝑓𝑥) / 𝑦]𝜑))
31, 2ac6gf 35128 . 2 ((𝐴𝑀 ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑))
4 fdm 6502 . . . . . . 7 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
5 vex 3472 . . . . . . . 8 𝑓 ∈ V
65dmex 7602 . . . . . . 7 dom 𝑓 ∈ V
74, 6eqeltrrdi 2923 . . . . . 6 (𝑓:𝐴𝐵𝐴 ∈ V)
8 ffn 6494 . . . . . 6 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
9 fnrndomg 9947 . . . . . 6 (𝐴 ∈ V → (𝑓 Fn 𝐴 → ran 𝑓𝐴))
107, 8, 9sylc 65 . . . . 5 (𝑓:𝐴𝐵 → ran 𝑓𝐴)
1110adantr 484 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ran 𝑓𝐴)
12 frn 6500 . . . . 5 (𝑓:𝐴𝐵 → ran 𝑓𝐵)
1312adantr 484 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ran 𝑓𝐵)
14 nfv 1915 . . . . . 6 𝑥 𝑓:𝐴𝐵
15 nfra1 3208 . . . . . 6 𝑥𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑
1614, 15nfan 1900 . . . . 5 𝑥(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)
17 ffun 6497 . . . . . . . . . 10 (𝑓:𝐴𝐵 → Fun 𝑓)
1817adantr 484 . . . . . . . . 9 ((𝑓:𝐴𝐵𝑥𝐴) → Fun 𝑓)
194eleq2d 2899 . . . . . . . . . 10 (𝑓:𝐴𝐵 → (𝑥 ∈ dom 𝑓𝑥𝐴))
2019biimpar 481 . . . . . . . . 9 ((𝑓:𝐴𝐵𝑥𝐴) → 𝑥 ∈ dom 𝑓)
21 fvelrn 6826 . . . . . . . . 9 ((Fun 𝑓𝑥 ∈ dom 𝑓) → (𝑓𝑥) ∈ ran 𝑓)
2218, 20, 21syl2anc 587 . . . . . . . 8 ((𝑓:𝐴𝐵𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
2322adantlr 714 . . . . . . 7 (((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) ∧ 𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
24 rspa 3196 . . . . . . . 8 ((∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑𝑥𝐴) → [(𝑓𝑥) / 𝑦]𝜑)
2524adantll 713 . . . . . . 7 (((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) ∧ 𝑥𝐴) → [(𝑓𝑥) / 𝑦]𝜑)
26 rspesbca 3837 . . . . . . 7 (((𝑓𝑥) ∈ ran 𝑓[(𝑓𝑥) / 𝑦]𝜑) → ∃𝑦 ∈ ran 𝑓𝜑)
2723, 25, 26syl2anc 587 . . . . . 6 (((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) ∧ 𝑥𝐴) → ∃𝑦 ∈ ran 𝑓𝜑)
2827ex 416 . . . . 5 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑥𝐴 → ∃𝑦 ∈ ran 𝑓𝜑))
2916, 28ralrimi 3205 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑)
30 nfv 1915 . . . . . 6 𝑦 𝑓:𝐴𝐵
31 nfcv 2979 . . . . . . 7 𝑦𝐴
3231, 1nfralw 3214 . . . . . 6 𝑦𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑
3330, 32nfan 1900 . . . . 5 𝑦(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)
34 fvelrnb 6708 . . . . . . . 8 (𝑓 Fn 𝐴 → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥𝐴 (𝑓𝑥) = 𝑦))
358, 34syl 17 . . . . . . 7 (𝑓:𝐴𝐵 → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥𝐴 (𝑓𝑥) = 𝑦))
3635adantr 484 . . . . . 6 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥𝐴 (𝑓𝑥) = 𝑦))
37 rsp 3195 . . . . . . . . 9 (∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑 → (𝑥𝐴[(𝑓𝑥) / 𝑦]𝜑))
3837adantl 485 . . . . . . . 8 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑥𝐴[(𝑓𝑥) / 𝑦]𝜑))
392eqcoms 2830 . . . . . . . . 9 ((𝑓𝑥) = 𝑦 → (𝜑[(𝑓𝑥) / 𝑦]𝜑))
4039biimprcd 253 . . . . . . . 8 ([(𝑓𝑥) / 𝑦]𝜑 → ((𝑓𝑥) = 𝑦𝜑))
4138, 40syl6 35 . . . . . . 7 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑥𝐴 → ((𝑓𝑥) = 𝑦𝜑)))
4216, 41reximdai 3297 . . . . . 6 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (∃𝑥𝐴 (𝑓𝑥) = 𝑦 → ∃𝑥𝐴 𝜑))
4336, 42sylbid 243 . . . . 5 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑦 ∈ ran 𝑓 → ∃𝑥𝐴 𝜑))
4433, 43ralrimi 3205 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)
455rnex 7603 . . . . 5 ran 𝑓 ∈ V
46 breq1 5045 . . . . . . 7 (𝑐 = ran 𝑓 → (𝑐𝐴 ↔ ran 𝑓𝐴))
47 sseq1 3967 . . . . . . 7 (𝑐 = ran 𝑓 → (𝑐𝐵 ↔ ran 𝑓𝐵))
4846, 47anbi12d 633 . . . . . 6 (𝑐 = ran 𝑓 → ((𝑐𝐴𝑐𝐵) ↔ (ran 𝑓𝐴 ∧ ran 𝑓𝐵)))
49 rexeq 3387 . . . . . . . 8 (𝑐 = ran 𝑓 → (∃𝑦𝑐 𝜑 ↔ ∃𝑦 ∈ ran 𝑓𝜑))
5049ralbidv 3187 . . . . . . 7 (𝑐 = ran 𝑓 → (∀𝑥𝐴𝑦𝑐 𝜑 ↔ ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑))
51 raleq 3386 . . . . . . 7 (𝑐 = ran 𝑓 → (∀𝑦𝑐𝑥𝐴 𝜑 ↔ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑))
5250, 51anbi12d 633 . . . . . 6 (𝑐 = ran 𝑓 → ((∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑) ↔ (∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)))
5348, 52anbi12d 633 . . . . 5 (𝑐 = ran 𝑓 → (((𝑐𝐴𝑐𝐵) ∧ (∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑)) ↔ ((ran 𝑓𝐴 ∧ ran 𝑓𝐵) ∧ (∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑))))
5445, 53spcev 3582 . . . 4 (((ran 𝑓𝐴 ∧ ran 𝑓𝐵) ∧ (∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)) → ∃𝑐((𝑐𝐴𝑐𝐵) ∧ (∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑)))
5511, 13, 29, 44, 54syl22anc 837 . . 3 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∃𝑐((𝑐𝐴𝑐𝐵) ∧ (∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑)))
5655exlimiv 1931 . 2 (∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∃𝑐((𝑐𝐴𝑐𝐵) ∧ (∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑)))
573, 56syl 17 1 ((𝐴𝑀 ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑐((𝑐𝐴𝑐𝐵) ∧ (∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2114  ∀wral 3130  ∃wrex 3131  Vcvv 3469  [wsbc 3747   ⊆ wss 3908   class class class wbr 5042  dom cdm 5532  ran crn 5533  Fun wfun 6328   Fn wfn 6329  ⟶wf 6330  ‘cfv 6334   ≼ cdom 8494 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-reg 9044  ax-inf2 9092  ax-ac2 9874 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-r1 9181  df-rank 9182  df-card 9356  df-acn 9359  df-ac 9531 This theorem is referenced by: (None)
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