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Theorem indexdom 36048
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 3754 . . 3 𝑦[(𝑓𝑥) / 𝑦]𝜑
2 sbceq1a 3745 . . 3 (𝑦 = (𝑓𝑥) → (𝜑[(𝑓𝑥) / 𝑦]𝜑))
31, 2ac6gf 36046 . 2 ((𝐴𝑀 ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑))
4 fdm 6669 . . . . . . 7 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
5 vex 3447 . . . . . . . 8 𝑓 ∈ V
65dmex 7835 . . . . . . 7 dom 𝑓 ∈ V
74, 6eqeltrrdi 2847 . . . . . 6 (𝑓:𝐴𝐵𝐴 ∈ V)
8 ffn 6660 . . . . . 6 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
9 fnrndomg 10402 . . . . . 6 (𝐴 ∈ V → (𝑓 Fn 𝐴 → ran 𝑓𝐴))
107, 8, 9sylc 65 . . . . 5 (𝑓:𝐴𝐵 → ran 𝑓𝐴)
1110adantr 482 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ran 𝑓𝐴)
12 frn 6667 . . . . 5 (𝑓:𝐴𝐵 → ran 𝑓𝐵)
1312adantr 482 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ran 𝑓𝐵)
14 nfv 1917 . . . . . 6 𝑥 𝑓:𝐴𝐵
15 nfra1 3265 . . . . . 6 𝑥𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑
1614, 15nfan 1902 . . . . 5 𝑥(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)
17 ffun 6663 . . . . . . . . . 10 (𝑓:𝐴𝐵 → Fun 𝑓)
1817adantr 482 . . . . . . . . 9 ((𝑓:𝐴𝐵𝑥𝐴) → Fun 𝑓)
194eleq2d 2823 . . . . . . . . . 10 (𝑓:𝐴𝐵 → (𝑥 ∈ dom 𝑓𝑥𝐴))
2019biimpar 479 . . . . . . . . 9 ((𝑓:𝐴𝐵𝑥𝐴) → 𝑥 ∈ dom 𝑓)
21 fvelrn 7019 . . . . . . . . 9 ((Fun 𝑓𝑥 ∈ dom 𝑓) → (𝑓𝑥) ∈ ran 𝑓)
2218, 20, 21syl2anc 585 . . . . . . . 8 ((𝑓:𝐴𝐵𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
2322adantlr 713 . . . . . . 7 (((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) ∧ 𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
24 rspa 3229 . . . . . . . 8 ((∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑𝑥𝐴) → [(𝑓𝑥) / 𝑦]𝜑)
2524adantll 712 . . . . . . 7 (((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) ∧ 𝑥𝐴) → [(𝑓𝑥) / 𝑦]𝜑)
26 rspesbca 3832 . . . . . . 7 (((𝑓𝑥) ∈ ran 𝑓[(𝑓𝑥) / 𝑦]𝜑) → ∃𝑦 ∈ ran 𝑓𝜑)
2723, 25, 26syl2anc 585 . . . . . 6 (((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) ∧ 𝑥𝐴) → ∃𝑦 ∈ ran 𝑓𝜑)
2827ex 414 . . . . 5 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑥𝐴 → ∃𝑦 ∈ ran 𝑓𝜑))
2916, 28ralrimi 3238 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑)
30 nfv 1917 . . . . . 6 𝑦 𝑓:𝐴𝐵
31 nfcv 2905 . . . . . . 7 𝑦𝐴
3231, 1nfralw 3292 . . . . . 6 𝑦𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑
3330, 32nfan 1902 . . . . 5 𝑦(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)
34 fvelrnb 6895 . . . . . . . 8 (𝑓 Fn 𝐴 → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥𝐴 (𝑓𝑥) = 𝑦))
358, 34syl 17 . . . . . . 7 (𝑓:𝐴𝐵 → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥𝐴 (𝑓𝑥) = 𝑦))
3635adantr 482 . . . . . 6 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥𝐴 (𝑓𝑥) = 𝑦))
37 rsp 3228 . . . . . . . . 9 (∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑 → (𝑥𝐴[(𝑓𝑥) / 𝑦]𝜑))
3837adantl 483 . . . . . . . 8 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑥𝐴[(𝑓𝑥) / 𝑦]𝜑))
392eqcoms 2745 . . . . . . . . 9 ((𝑓𝑥) = 𝑦 → (𝜑[(𝑓𝑥) / 𝑦]𝜑))
4039biimprcd 250 . . . . . . . 8 ([(𝑓𝑥) / 𝑦]𝜑 → ((𝑓𝑥) = 𝑦𝜑))
4138, 40syl6 35 . . . . . . 7 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑥𝐴 → ((𝑓𝑥) = 𝑦𝜑)))
4216, 41reximdai 3242 . . . . . 6 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (∃𝑥𝐴 (𝑓𝑥) = 𝑦 → ∃𝑥𝐴 𝜑))
4336, 42sylbid 239 . . . . 5 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑦 ∈ ran 𝑓 → ∃𝑥𝐴 𝜑))
4433, 43ralrimi 3238 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)
455rnex 7836 . . . . 5 ran 𝑓 ∈ V
46 breq1 5103 . . . . . . 7 (𝑐 = ran 𝑓 → (𝑐𝐴 ↔ ran 𝑓𝐴))
47 sseq1 3964 . . . . . . 7 (𝑐 = ran 𝑓 → (𝑐𝐵 ↔ ran 𝑓𝐵))
4846, 47anbi12d 632 . . . . . 6 (𝑐 = ran 𝑓 → ((𝑐𝐴𝑐𝐵) ↔ (ran 𝑓𝐴 ∧ ran 𝑓𝐵)))
49 rexeq 3308 . . . . . . . 8 (𝑐 = ran 𝑓 → (∃𝑦𝑐 𝜑 ↔ ∃𝑦 ∈ ran 𝑓𝜑))
5049ralbidv 3172 . . . . . . 7 (𝑐 = ran 𝑓 → (∀𝑥𝐴𝑦𝑐 𝜑 ↔ ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑))
51 raleq 3307 . . . . . . 7 (𝑐 = ran 𝑓 → (∀𝑦𝑐𝑥𝐴 𝜑 ↔ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑))
5250, 51anbi12d 632 . . . . . 6 (𝑐 = ran 𝑓 → ((∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑) ↔ (∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)))
5348, 52anbi12d 632 . . . . 5 (𝑐 = ran 𝑓 → (((𝑐𝐴𝑐𝐵) ∧ (∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑)) ↔ ((ran 𝑓𝐴 ∧ ran 𝑓𝐵) ∧ (∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑))))
5445, 53spcev 3560 . . . 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 1933 . 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 205  wa 397   = wceq 1541  wex 1781  wcel 2106  wral 3062  wrex 3071  Vcvv 3443  [wsbc 3734  wss 3905   class class class wbr 5100  dom cdm 5627  ran crn 5628  Fun wfun 6482   Fn wfn 6483  wf 6484  cfv 6488  cdom 8811
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5237  ax-sep 5251  ax-nul 5258  ax-pow 5315  ax-pr 5379  ax-un 7659  ax-reg 9458  ax-inf2 9507  ax-ac2 10329
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3735  df-csb 3851  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3924  df-nul 4278  df-if 4482  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4861  df-int 4903  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5184  df-tr 5218  df-id 5525  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5582  df-se 5583  df-we 5584  df-xp 5633  df-rel 5634  df-cnv 5635  df-co 5636  df-dm 5637  df-rn 5638  df-res 5639  df-ima 5640  df-pred 6246  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6440  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-isom 6497  df-riota 7302  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7790  df-1st 7908  df-2nd 7909  df-frecs 8176  df-wrecs 8207  df-recs 8281  df-rdg 8320  df-er 8578  df-map 8697  df-en 8814  df-dom 8815  df-r1 9630  df-rank 9631  df-card 9805  df-acn 9808  df-ac 9982
This theorem is referenced by: (None)
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