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Theorem indexdom 36597
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 3797 . . 3 𝑦[(𝑓𝑥) / 𝑦]𝜑
2 sbceq1a 3788 . . 3 (𝑦 = (𝑓𝑥) → (𝜑[(𝑓𝑥) / 𝑦]𝜑))
31, 2ac6gf 36595 . 2 ((𝐴𝑀 ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑))
4 fdm 6726 . . . . . . 7 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
5 vex 3478 . . . . . . . 8 𝑓 ∈ V
65dmex 7901 . . . . . . 7 dom 𝑓 ∈ V
74, 6eqeltrrdi 2842 . . . . . 6 (𝑓:𝐴𝐵𝐴 ∈ V)
8 ffn 6717 . . . . . 6 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
9 fnrndomg 10530 . . . . . 6 (𝐴 ∈ V → (𝑓 Fn 𝐴 → ran 𝑓𝐴))
107, 8, 9sylc 65 . . . . 5 (𝑓:𝐴𝐵 → ran 𝑓𝐴)
1110adantr 481 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ran 𝑓𝐴)
12 frn 6724 . . . . 5 (𝑓:𝐴𝐵 → ran 𝑓𝐵)
1312adantr 481 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ran 𝑓𝐵)
14 nfv 1917 . . . . . 6 𝑥 𝑓:𝐴𝐵
15 nfra1 3281 . . . . . 6 𝑥𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑
1614, 15nfan 1902 . . . . 5 𝑥(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)
17 ffun 6720 . . . . . . . . . 10 (𝑓:𝐴𝐵 → Fun 𝑓)
1817adantr 481 . . . . . . . . 9 ((𝑓:𝐴𝐵𝑥𝐴) → Fun 𝑓)
194eleq2d 2819 . . . . . . . . . 10 (𝑓:𝐴𝐵 → (𝑥 ∈ dom 𝑓𝑥𝐴))
2019biimpar 478 . . . . . . . . 9 ((𝑓:𝐴𝐵𝑥𝐴) → 𝑥 ∈ dom 𝑓)
21 fvelrn 7078 . . . . . . . . 9 ((Fun 𝑓𝑥 ∈ dom 𝑓) → (𝑓𝑥) ∈ ran 𝑓)
2218, 20, 21syl2anc 584 . . . . . . . 8 ((𝑓:𝐴𝐵𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
2322adantlr 713 . . . . . . 7 (((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) ∧ 𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
24 rspa 3245 . . . . . . . 8 ((∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑𝑥𝐴) → [(𝑓𝑥) / 𝑦]𝜑)
2524adantll 712 . . . . . . 7 (((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) ∧ 𝑥𝐴) → [(𝑓𝑥) / 𝑦]𝜑)
26 rspesbca 3875 . . . . . . 7 (((𝑓𝑥) ∈ ran 𝑓[(𝑓𝑥) / 𝑦]𝜑) → ∃𝑦 ∈ ran 𝑓𝜑)
2723, 25, 26syl2anc 584 . . . . . 6 (((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) ∧ 𝑥𝐴) → ∃𝑦 ∈ ran 𝑓𝜑)
2827ex 413 . . . . 5 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑥𝐴 → ∃𝑦 ∈ ran 𝑓𝜑))
2916, 28ralrimi 3254 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑)
30 nfv 1917 . . . . . 6 𝑦 𝑓:𝐴𝐵
31 nfcv 2903 . . . . . . 7 𝑦𝐴
3231, 1nfralw 3308 . . . . . 6 𝑦𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑
3330, 32nfan 1902 . . . . 5 𝑦(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)
34 fvelrnb 6952 . . . . . . . 8 (𝑓 Fn 𝐴 → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥𝐴 (𝑓𝑥) = 𝑦))
358, 34syl 17 . . . . . . 7 (𝑓:𝐴𝐵 → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥𝐴 (𝑓𝑥) = 𝑦))
3635adantr 481 . . . . . 6 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥𝐴 (𝑓𝑥) = 𝑦))
37 rsp 3244 . . . . . . . . 9 (∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑 → (𝑥𝐴[(𝑓𝑥) / 𝑦]𝜑))
3837adantl 482 . . . . . . . 8 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑥𝐴[(𝑓𝑥) / 𝑦]𝜑))
392eqcoms 2740 . . . . . . . . 9 ((𝑓𝑥) = 𝑦 → (𝜑[(𝑓𝑥) / 𝑦]𝜑))
4039biimprcd 249 . . . . . . . 8 ([(𝑓𝑥) / 𝑦]𝜑 → ((𝑓𝑥) = 𝑦𝜑))
4138, 40syl6 35 . . . . . . 7 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑥𝐴 → ((𝑓𝑥) = 𝑦𝜑)))
4216, 41reximdai 3258 . . . . . 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 3254 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)
455rnex 7902 . . . . 5 ran 𝑓 ∈ V
46 breq1 5151 . . . . . . 7 (𝑐 = ran 𝑓 → (𝑐𝐴 ↔ ran 𝑓𝐴))
47 sseq1 4007 . . . . . . 7 (𝑐 = ran 𝑓 → (𝑐𝐵 ↔ ran 𝑓𝐵))
4846, 47anbi12d 631 . . . . . 6 (𝑐 = ran 𝑓 → ((𝑐𝐴𝑐𝐵) ↔ (ran 𝑓𝐴 ∧ ran 𝑓𝐵)))
49 rexeq 3321 . . . . . . . 8 (𝑐 = ran 𝑓 → (∃𝑦𝑐 𝜑 ↔ ∃𝑦 ∈ ran 𝑓𝜑))
5049ralbidv 3177 . . . . . . 7 (𝑐 = ran 𝑓 → (∀𝑥𝐴𝑦𝑐 𝜑 ↔ ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑))
51 raleq 3322 . . . . . . 7 (𝑐 = ran 𝑓 → (∀𝑦𝑐𝑥𝐴 𝜑 ↔ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑))
5250, 51anbi12d 631 . . . . . 6 (𝑐 = ran 𝑓 → ((∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑) ↔ (∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)))
5348, 52anbi12d 631 . . . . 5 (𝑐 = ran 𝑓 → (((𝑐𝐴𝑐𝐵) ∧ (∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑)) ↔ ((ran 𝑓𝐴 ∧ ran 𝑓𝐵) ∧ (∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑))))
5445, 53spcev 3596 . . . 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 396   = wceq 1541  wex 1781  wcel 2106  wral 3061  wrex 3070  Vcvv 3474  [wsbc 3777  wss 3948   class class class wbr 5148  dom cdm 5676  ran crn 5677  Fun wfun 6537   Fn wfn 6538  wf 6539  cfv 6543  cdom 8936
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-reg 9586  ax-inf2 9635  ax-ac2 10457
This theorem depends on definitions:  df-bi 206  df-an 397  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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-r1 9758  df-rank 9759  df-card 9933  df-acn 9936  df-ac 10110
This theorem is referenced by: (None)
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