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Theorem indexdom 37721
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 3811 . . 3 𝑦[(𝑓𝑥) / 𝑦]𝜑
2 sbceq1a 3802 . . 3 (𝑦 = (𝑓𝑥) → (𝜑[(𝑓𝑥) / 𝑦]𝜑))
31, 2ac6gf 37719 . 2 ((𝐴𝑀 ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑))
4 fdm 6746 . . . . . . 7 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
5 vex 3482 . . . . . . . 8 𝑓 ∈ V
65dmex 7932 . . . . . . 7 dom 𝑓 ∈ V
74, 6eqeltrrdi 2848 . . . . . 6 (𝑓:𝐴𝐵𝐴 ∈ V)
8 ffn 6737 . . . . . 6 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
9 fnrndomg 10574 . . . . . 6 (𝐴 ∈ V → (𝑓 Fn 𝐴 → ran 𝑓𝐴))
107, 8, 9sylc 65 . . . . 5 (𝑓:𝐴𝐵 → ran 𝑓𝐴)
1110adantr 480 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ran 𝑓𝐴)
12 frn 6744 . . . . 5 (𝑓:𝐴𝐵 → ran 𝑓𝐵)
1312adantr 480 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ran 𝑓𝐵)
14 nfv 1912 . . . . . 6 𝑥 𝑓:𝐴𝐵
15 nfra1 3282 . . . . . 6 𝑥𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑
1614, 15nfan 1897 . . . . 5 𝑥(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)
17 ffun 6740 . . . . . . . . . 10 (𝑓:𝐴𝐵 → Fun 𝑓)
1817adantr 480 . . . . . . . . 9 ((𝑓:𝐴𝐵𝑥𝐴) → Fun 𝑓)
194eleq2d 2825 . . . . . . . . . 10 (𝑓:𝐴𝐵 → (𝑥 ∈ dom 𝑓𝑥𝐴))
2019biimpar 477 . . . . . . . . 9 ((𝑓:𝐴𝐵𝑥𝐴) → 𝑥 ∈ dom 𝑓)
21 fvelrn 7096 . . . . . . . . 9 ((Fun 𝑓𝑥 ∈ dom 𝑓) → (𝑓𝑥) ∈ ran 𝑓)
2218, 20, 21syl2anc 584 . . . . . . . 8 ((𝑓:𝐴𝐵𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
2322adantlr 715 . . . . . . 7 (((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) ∧ 𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
24 rspa 3246 . . . . . . . 8 ((∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑𝑥𝐴) → [(𝑓𝑥) / 𝑦]𝜑)
2524adantll 714 . . . . . . 7 (((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) ∧ 𝑥𝐴) → [(𝑓𝑥) / 𝑦]𝜑)
26 rspesbca 3890 . . . . . . 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 412 . . . . 5 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑥𝐴 → ∃𝑦 ∈ ran 𝑓𝜑))
2916, 28ralrimi 3255 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑)
30 nfv 1912 . . . . . 6 𝑦 𝑓:𝐴𝐵
31 nfcv 2903 . . . . . . 7 𝑦𝐴
3231, 1nfralw 3309 . . . . . 6 𝑦𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑
3330, 32nfan 1897 . . . . 5 𝑦(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)
34 fvelrnb 6969 . . . . . . . 8 (𝑓 Fn 𝐴 → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥𝐴 (𝑓𝑥) = 𝑦))
358, 34syl 17 . . . . . . 7 (𝑓:𝐴𝐵 → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥𝐴 (𝑓𝑥) = 𝑦))
3635adantr 480 . . . . . 6 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥𝐴 (𝑓𝑥) = 𝑦))
37 rsp 3245 . . . . . . . . 9 (∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑 → (𝑥𝐴[(𝑓𝑥) / 𝑦]𝜑))
3837adantl 481 . . . . . . . 8 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑥𝐴[(𝑓𝑥) / 𝑦]𝜑))
392eqcoms 2743 . . . . . . . . 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 3259 . . . . . 6 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (∃𝑥𝐴 (𝑓𝑥) = 𝑦 → ∃𝑥𝐴 𝜑))
4336, 42sylbid 240 . . . . 5 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑦 ∈ ran 𝑓 → ∃𝑥𝐴 𝜑))
4433, 43ralrimi 3255 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)
455rnex 7933 . . . . 5 ran 𝑓 ∈ V
46 breq1 5151 . . . . . . 7 (𝑐 = ran 𝑓 → (𝑐𝐴 ↔ ran 𝑓𝐴))
47 sseq1 4021 . . . . . . 7 (𝑐 = ran 𝑓 → (𝑐𝐵 ↔ ran 𝑓𝐵))
4846, 47anbi12d 632 . . . . . 6 (𝑐 = ran 𝑓 → ((𝑐𝐴𝑐𝐵) ↔ (ran 𝑓𝐴 ∧ ran 𝑓𝐵)))
49 rexeq 3320 . . . . . . . 8 (𝑐 = ran 𝑓 → (∃𝑦𝑐 𝜑 ↔ ∃𝑦 ∈ ran 𝑓𝜑))
5049ralbidv 3176 . . . . . . 7 (𝑐 = ran 𝑓 → (∀𝑥𝐴𝑦𝑐 𝜑 ↔ ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑))
51 raleq 3321 . . . . . . 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 3606 . . . 4 (((ran 𝑓𝐴 ∧ ran 𝑓𝐵) ∧ (∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)) → ∃𝑐((𝑐𝐴𝑐𝐵) ∧ (∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑)))
5511, 13, 29, 44, 54syl22anc 839 . . 3 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∃𝑐((𝑐𝐴𝑐𝐵) ∧ (∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑)))
5655exlimiv 1928 . 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 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  [wsbc 3791  wss 3963   class class class wbr 5148  dom cdm 5689  ran crn 5690  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  cdom 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-r1 9802  df-rank 9803  df-card 9977  df-acn 9980  df-ac 10154
This theorem is referenced by: (None)
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