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Theorem indexdom 36905
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 3796 . . 3 𝑦[(𝑓𝑥) / 𝑦]𝜑
2 sbceq1a 3787 . . 3 (𝑦 = (𝑓𝑥) → (𝜑[(𝑓𝑥) / 𝑦]𝜑))
31, 2ac6gf 36903 . 2 ((𝐴𝑀 ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑))
4 fdm 6725 . . . . . . 7 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
5 vex 3476 . . . . . . . 8 𝑓 ∈ V
65dmex 7904 . . . . . . 7 dom 𝑓 ∈ V
74, 6eqeltrrdi 2840 . . . . . 6 (𝑓:𝐴𝐵𝐴 ∈ V)
8 ffn 6716 . . . . . 6 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
9 fnrndomg 10533 . . . . . 6 (𝐴 ∈ V → (𝑓 Fn 𝐴 → ran 𝑓𝐴))
107, 8, 9sylc 65 . . . . 5 (𝑓:𝐴𝐵 → ran 𝑓𝐴)
1110adantr 479 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ran 𝑓𝐴)
12 frn 6723 . . . . 5 (𝑓:𝐴𝐵 → ran 𝑓𝐵)
1312adantr 479 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ran 𝑓𝐵)
14 nfv 1915 . . . . . 6 𝑥 𝑓:𝐴𝐵
15 nfra1 3279 . . . . . 6 𝑥𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑
1614, 15nfan 1900 . . . . 5 𝑥(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)
17 ffun 6719 . . . . . . . . . 10 (𝑓:𝐴𝐵 → Fun 𝑓)
1817adantr 479 . . . . . . . . 9 ((𝑓:𝐴𝐵𝑥𝐴) → Fun 𝑓)
194eleq2d 2817 . . . . . . . . . 10 (𝑓:𝐴𝐵 → (𝑥 ∈ dom 𝑓𝑥𝐴))
2019biimpar 476 . . . . . . . . 9 ((𝑓:𝐴𝐵𝑥𝐴) → 𝑥 ∈ dom 𝑓)
21 fvelrn 7077 . . . . . . . . 9 ((Fun 𝑓𝑥 ∈ dom 𝑓) → (𝑓𝑥) ∈ ran 𝑓)
2218, 20, 21syl2anc 582 . . . . . . . 8 ((𝑓:𝐴𝐵𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
2322adantlr 711 . . . . . . 7 (((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) ∧ 𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
24 rspa 3243 . . . . . . . 8 ((∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑𝑥𝐴) → [(𝑓𝑥) / 𝑦]𝜑)
2524adantll 710 . . . . . . 7 (((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) ∧ 𝑥𝐴) → [(𝑓𝑥) / 𝑦]𝜑)
26 rspesbca 3874 . . . . . . 7 (((𝑓𝑥) ∈ ran 𝑓[(𝑓𝑥) / 𝑦]𝜑) → ∃𝑦 ∈ ran 𝑓𝜑)
2723, 25, 26syl2anc 582 . . . . . 6 (((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) ∧ 𝑥𝐴) → ∃𝑦 ∈ ran 𝑓𝜑)
2827ex 411 . . . . 5 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑥𝐴 → ∃𝑦 ∈ ran 𝑓𝜑))
2916, 28ralrimi 3252 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑)
30 nfv 1915 . . . . . 6 𝑦 𝑓:𝐴𝐵
31 nfcv 2901 . . . . . . 7 𝑦𝐴
3231, 1nfralw 3306 . . . . . 6 𝑦𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑
3330, 32nfan 1900 . . . . 5 𝑦(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)
34 fvelrnb 6951 . . . . . . . 8 (𝑓 Fn 𝐴 → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥𝐴 (𝑓𝑥) = 𝑦))
358, 34syl 17 . . . . . . 7 (𝑓:𝐴𝐵 → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥𝐴 (𝑓𝑥) = 𝑦))
3635adantr 479 . . . . . 6 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥𝐴 (𝑓𝑥) = 𝑦))
37 rsp 3242 . . . . . . . . 9 (∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑 → (𝑥𝐴[(𝑓𝑥) / 𝑦]𝜑))
3837adantl 480 . . . . . . . 8 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → (𝑥𝐴[(𝑓𝑥) / 𝑦]𝜑))
392eqcoms 2738 . . . . . . . . 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 3256 . . . . . 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 3252 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)
455rnex 7905 . . . . 5 ran 𝑓 ∈ V
46 breq1 5150 . . . . . . 7 (𝑐 = ran 𝑓 → (𝑐𝐴 ↔ ran 𝑓𝐴))
47 sseq1 4006 . . . . . . 7 (𝑐 = ran 𝑓 → (𝑐𝐵 ↔ ran 𝑓𝐵))
4846, 47anbi12d 629 . . . . . 6 (𝑐 = ran 𝑓 → ((𝑐𝐴𝑐𝐵) ↔ (ran 𝑓𝐴 ∧ ran 𝑓𝐵)))
49 rexeq 3319 . . . . . . . 8 (𝑐 = ran 𝑓 → (∃𝑦𝑐 𝜑 ↔ ∃𝑦 ∈ ran 𝑓𝜑))
5049ralbidv 3175 . . . . . . 7 (𝑐 = ran 𝑓 → (∀𝑥𝐴𝑦𝑐 𝜑 ↔ ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑))
51 raleq 3320 . . . . . . 7 (𝑐 = ran 𝑓 → (∀𝑦𝑐𝑥𝐴 𝜑 ↔ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑))
5250, 51anbi12d 629 . . . . . 6 (𝑐 = ran 𝑓 → ((∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑) ↔ (∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)))
5348, 52anbi12d 629 . . . . 5 (𝑐 = ran 𝑓 → (((𝑐𝐴𝑐𝐵) ∧ (∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑)) ↔ ((ran 𝑓𝐴 ∧ ran 𝑓𝐵) ∧ (∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑))))
5445, 53spcev 3595 . . . 4 (((ran 𝑓𝐴 ∧ ran 𝑓𝐵) ∧ (∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)) → ∃𝑐((𝑐𝐴𝑐𝐵) ∧ (∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑)))
5511, 13, 29, 44, 54syl22anc 835 . . 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 205  wa 394   = wceq 1539  wex 1779  wcel 2104  wral 3059  wrex 3068  Vcvv 3472  [wsbc 3776  wss 3947   class class class wbr 5147  dom cdm 5675  ran crn 5676  Fun wfun 6536   Fn wfn 6537  wf 6538  cfv 6542  cdom 8939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-reg 9589  ax-inf2 9638  ax-ac2 10460
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-r1 9761  df-rank 9762  df-card 9936  df-acn 9939  df-ac 10113
This theorem is referenced by: (None)
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