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Theorem indexfi 8824
Description: If for every element of a finite indexing set 𝐴 there exists a corresponding element of another set 𝐵, then there exists a finite subset of 𝐵 consisting only of those elements which are indexed by 𝐴. Proven without the Axiom of Choice, unlike indexdom 34877. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
indexfi ((𝐴 ∈ Fin ∧ 𝐵𝑀 ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑐 ∈ Fin (𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑))
Distinct variable groups:   𝑥,𝑐,𝑦,𝐴   𝐵,𝑐,𝑥,𝑦   𝜑,𝑐
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑀(𝑥,𝑦,𝑐)

Proof of Theorem indexfi
Dummy variables 𝑓 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1908 . . . . . 6 𝑧𝜑
2 nfsbc1v 3795 . . . . . 6 𝑦[𝑧 / 𝑦]𝜑
3 sbceq1a 3786 . . . . . 6 (𝑦 = 𝑧 → (𝜑[𝑧 / 𝑦]𝜑))
41, 2, 3cbvrexw 3447 . . . . 5 (∃𝑦𝐵 𝜑 ↔ ∃𝑧𝐵 [𝑧 / 𝑦]𝜑)
54ralbii 3169 . . . 4 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑧𝐵 [𝑧 / 𝑦]𝜑)
6 dfsbcq 3777 . . . . 5 (𝑧 = (𝑓𝑥) → ([𝑧 / 𝑦]𝜑[(𝑓𝑥) / 𝑦]𝜑))
76ac6sfi 8754 . . . 4 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑧𝐵 [𝑧 / 𝑦]𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑))
85, 7sylan2b 593 . . 3 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑))
9 simpll 763 . . . . 5 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → 𝐴 ∈ Fin)
10 ffn 6510 . . . . . . 7 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
1110ad2antrl 724 . . . . . 6 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → 𝑓 Fn 𝐴)
12 dffn4 6592 . . . . . 6 (𝑓 Fn 𝐴𝑓:𝐴onto→ran 𝑓)
1311, 12sylib 219 . . . . 5 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → 𝑓:𝐴onto→ran 𝑓)
14 fofi 8802 . . . . 5 ((𝐴 ∈ Fin ∧ 𝑓:𝐴onto→ran 𝑓) → ran 𝑓 ∈ Fin)
159, 13, 14syl2anc 584 . . . 4 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ran 𝑓 ∈ Fin)
16 frn 6516 . . . . 5 (𝑓:𝐴𝐵 → ran 𝑓𝐵)
1716ad2antrl 724 . . . 4 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ran 𝑓𝐵)
18 fnfvelrn 6843 . . . . . . . . 9 ((𝑓 Fn 𝐴𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
1910, 18sylan 580 . . . . . . . 8 ((𝑓:𝐴𝐵𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
20 rspesbca 3867 . . . . . . . . 9 (((𝑓𝑥) ∈ ran 𝑓[(𝑓𝑥) / 𝑦]𝜑) → ∃𝑦 ∈ ran 𝑓𝜑)
2120ex 413 . . . . . . . 8 ((𝑓𝑥) ∈ ran 𝑓 → ([(𝑓𝑥) / 𝑦]𝜑 → ∃𝑦 ∈ ran 𝑓𝜑))
2219, 21syl 17 . . . . . . 7 ((𝑓:𝐴𝐵𝑥𝐴) → ([(𝑓𝑥) / 𝑦]𝜑 → ∃𝑦 ∈ ran 𝑓𝜑))
2322ralimdva 3181 . . . . . 6 (𝑓:𝐴𝐵 → (∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑 → ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑))
2423imp 407 . . . . 5 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑)
2524adantl 482 . . . 4 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑)
26 simpr 485 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) ∧ 𝑤𝐴) → 𝑤𝐴)
27 simprr 769 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)
28 nfv 1908 . . . . . . . . . . 11 𝑤[(𝑓𝑥) / 𝑦]𝜑
29 nfsbc1v 3795 . . . . . . . . . . 11 𝑥[𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑
30 fveq2 6666 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝑓𝑥) = (𝑓𝑤))
3130sbceq1d 3780 . . . . . . . . . . . 12 (𝑥 = 𝑤 → ([(𝑓𝑥) / 𝑦]𝜑[(𝑓𝑤) / 𝑦]𝜑))
32 sbceq1a 3786 . . . . . . . . . . . 12 (𝑥 = 𝑤 → ([(𝑓𝑤) / 𝑦]𝜑[𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑))
3331, 32bitrd 280 . . . . . . . . . . 11 (𝑥 = 𝑤 → ([(𝑓𝑥) / 𝑦]𝜑[𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑))
3428, 29, 33cbvralw 3446 . . . . . . . . . 10 (∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑 ↔ ∀𝑤𝐴 [𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑)
3527, 34sylib 219 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∀𝑤𝐴 [𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑)
3635r19.21bi 3212 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) ∧ 𝑤𝐴) → [𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑)
37 rspesbca 3867 . . . . . . . 8 ((𝑤𝐴[𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑) → ∃𝑥𝐴 [(𝑓𝑤) / 𝑦]𝜑)
3826, 36, 37syl2anc 584 . . . . . . 7 ((((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) ∧ 𝑤𝐴) → ∃𝑥𝐴 [(𝑓𝑤) / 𝑦]𝜑)
3938ralrimiva 3186 . . . . . 6 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∀𝑤𝐴𝑥𝐴 [(𝑓𝑤) / 𝑦]𝜑)
40 dfsbcq 3777 . . . . . . . . 9 (𝑧 = (𝑓𝑤) → ([𝑧 / 𝑦]𝜑[(𝑓𝑤) / 𝑦]𝜑))
4140rexbidv 3301 . . . . . . . 8 (𝑧 = (𝑓𝑤) → (∃𝑥𝐴 [𝑧 / 𝑦]𝜑 ↔ ∃𝑥𝐴 [(𝑓𝑤) / 𝑦]𝜑))
4241ralrn 6849 . . . . . . 7 (𝑓 Fn 𝐴 → (∀𝑧 ∈ ran 𝑓𝑥𝐴 [𝑧 / 𝑦]𝜑 ↔ ∀𝑤𝐴𝑥𝐴 [(𝑓𝑤) / 𝑦]𝜑))
4311, 42syl 17 . . . . . 6 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → (∀𝑧 ∈ ran 𝑓𝑥𝐴 [𝑧 / 𝑦]𝜑 ↔ ∀𝑤𝐴𝑥𝐴 [(𝑓𝑤) / 𝑦]𝜑))
4439, 43mpbird 258 . . . . 5 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∀𝑧 ∈ ran 𝑓𝑥𝐴 [𝑧 / 𝑦]𝜑)
45 nfv 1908 . . . . . 6 𝑧𝑥𝐴 𝜑
46 nfcv 2981 . . . . . . 7 𝑦𝐴
4746, 2nfrex 3313 . . . . . 6 𝑦𝑥𝐴 [𝑧 / 𝑦]𝜑
483rexbidv 3301 . . . . . 6 (𝑦 = 𝑧 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑))
4945, 47, 48cbvralw 3446 . . . . 5 (∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑 ↔ ∀𝑧 ∈ ran 𝑓𝑥𝐴 [𝑧 / 𝑦]𝜑)
5044, 49sylibr 235 . . . 4 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)
51 sseq1 3995 . . . . . 6 (𝑐 = ran 𝑓 → (𝑐𝐵 ↔ ran 𝑓𝐵))
52 rexeq 3411 . . . . . . 7 (𝑐 = ran 𝑓 → (∃𝑦𝑐 𝜑 ↔ ∃𝑦 ∈ ran 𝑓𝜑))
5352ralbidv 3201 . . . . . 6 (𝑐 = ran 𝑓 → (∀𝑥𝐴𝑦𝑐 𝜑 ↔ ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑))
54 raleq 3410 . . . . . 6 (𝑐 = ran 𝑓 → (∀𝑦𝑐𝑥𝐴 𝜑 ↔ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑))
5551, 53, 543anbi123d 1429 . . . . 5 (𝑐 = ran 𝑓 → ((𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑) ↔ (ran 𝑓𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)))
5655rspcev 3626 . . . 4 ((ran 𝑓 ∈ Fin ∧ (ran 𝑓𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)) → ∃𝑐 ∈ Fin (𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑))
5715, 17, 25, 50, 56syl13anc 1366 . . 3 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∃𝑐 ∈ Fin (𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑))
588, 57exlimddv 1929 . 2 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑐 ∈ Fin (𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑))
59583adant2 1125 1 ((𝐴 ∈ Fin ∧ 𝐵𝑀 ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑐 ∈ Fin (𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  wral 3142  wrex 3143  [wsbc 3775  wss 3939  ran crn 5554   Fn wfn 6346  wf 6347  ontowfo 6349  cfv 6351  Fincfn 8501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-om 7572  df-1o 8096  df-er 8282  df-en 8502  df-dom 8503  df-fin 8505
This theorem is referenced by:  filbcmb  34883
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