MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  indexfi Structured version   Visualization version   GIF version

Theorem indexfi 8315
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 33659. (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 1883 . . . . . 6 𝑧𝜑
2 nfsbc1v 3488 . . . . . 6 𝑦[𝑧 / 𝑦]𝜑
3 sbceq1a 3479 . . . . . 6 (𝑦 = 𝑧 → (𝜑[𝑧 / 𝑦]𝜑))
41, 2, 3cbvrex 3198 . . . . 5 (∃𝑦𝐵 𝜑 ↔ ∃𝑧𝐵 [𝑧 / 𝑦]𝜑)
54ralbii 3009 . . . 4 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑧𝐵 [𝑧 / 𝑦]𝜑)
6 dfsbcq 3470 . . . . 5 (𝑧 = (𝑓𝑥) → ([𝑧 / 𝑦]𝜑[(𝑓𝑥) / 𝑦]𝜑))
76ac6sfi 8245 . . . 4 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑧𝐵 [𝑧 / 𝑦]𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑))
85, 7sylan2b 491 . . 3 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑))
9 simpll 805 . . . . 5 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → 𝐴 ∈ Fin)
10 ffn 6083 . . . . . . 7 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
1110ad2antrl 764 . . . . . 6 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → 𝑓 Fn 𝐴)
12 dffn4 6159 . . . . . 6 (𝑓 Fn 𝐴𝑓:𝐴onto→ran 𝑓)
1311, 12sylib 208 . . . . 5 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → 𝑓:𝐴onto→ran 𝑓)
14 fofi 8293 . . . . 5 ((𝐴 ∈ Fin ∧ 𝑓:𝐴onto→ran 𝑓) → ran 𝑓 ∈ Fin)
159, 13, 14syl2anc 694 . . . 4 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ran 𝑓 ∈ Fin)
16 frn 6091 . . . . 5 (𝑓:𝐴𝐵 → ran 𝑓𝐵)
1716ad2antrl 764 . . . 4 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ran 𝑓𝐵)
18 fnfvelrn 6396 . . . . . . . . 9 ((𝑓 Fn 𝐴𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
1910, 18sylan 487 . . . . . . . 8 ((𝑓:𝐴𝐵𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
20 rspesbca 3553 . . . . . . . . 9 (((𝑓𝑥) ∈ ran 𝑓[(𝑓𝑥) / 𝑦]𝜑) → ∃𝑦 ∈ ran 𝑓𝜑)
2120ex 449 . . . . . . . 8 ((𝑓𝑥) ∈ ran 𝑓 → ([(𝑓𝑥) / 𝑦]𝜑 → ∃𝑦 ∈ ran 𝑓𝜑))
2219, 21syl 17 . . . . . . 7 ((𝑓:𝐴𝐵𝑥𝐴) → ([(𝑓𝑥) / 𝑦]𝜑 → ∃𝑦 ∈ ran 𝑓𝜑))
2322ralimdva 2991 . . . . . 6 (𝑓:𝐴𝐵 → (∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑 → ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑))
2423imp 444 . . . . 5 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑)
2524adantl 481 . . . 4 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑)
26 simpr 476 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) ∧ 𝑤𝐴) → 𝑤𝐴)
27 simprr 811 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)
28 nfv 1883 . . . . . . . . . . 11 𝑤[(𝑓𝑥) / 𝑦]𝜑
29 nfsbc1v 3488 . . . . . . . . . . 11 𝑥[𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑
30 fveq2 6229 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝑓𝑥) = (𝑓𝑤))
3130sbceq1d 3473 . . . . . . . . . . . 12 (𝑥 = 𝑤 → ([(𝑓𝑥) / 𝑦]𝜑[(𝑓𝑤) / 𝑦]𝜑))
32 sbceq1a 3479 . . . . . . . . . . . 12 (𝑥 = 𝑤 → ([(𝑓𝑤) / 𝑦]𝜑[𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑))
3331, 32bitrd 268 . . . . . . . . . . 11 (𝑥 = 𝑤 → ([(𝑓𝑥) / 𝑦]𝜑[𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑))
3428, 29, 33cbvral 3197 . . . . . . . . . 10 (∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑 ↔ ∀𝑤𝐴 [𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑)
3527, 34sylib 208 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∀𝑤𝐴 [𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑)
3635r19.21bi 2961 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) ∧ 𝑤𝐴) → [𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑)
37 rspesbca 3553 . . . . . . . 8 ((𝑤𝐴[𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑) → ∃𝑥𝐴 [(𝑓𝑤) / 𝑦]𝜑)
3826, 36, 37syl2anc 694 . . . . . . 7 ((((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) ∧ 𝑤𝐴) → ∃𝑥𝐴 [(𝑓𝑤) / 𝑦]𝜑)
3938ralrimiva 2995 . . . . . 6 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∀𝑤𝐴𝑥𝐴 [(𝑓𝑤) / 𝑦]𝜑)
40 dfsbcq 3470 . . . . . . . . 9 (𝑧 = (𝑓𝑤) → ([𝑧 / 𝑦]𝜑[(𝑓𝑤) / 𝑦]𝜑))
4140rexbidv 3081 . . . . . . . 8 (𝑧 = (𝑓𝑤) → (∃𝑥𝐴 [𝑧 / 𝑦]𝜑 ↔ ∃𝑥𝐴 [(𝑓𝑤) / 𝑦]𝜑))
4241ralrn 6402 . . . . . . 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 247 . . . . 5 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∀𝑧 ∈ ran 𝑓𝑥𝐴 [𝑧 / 𝑦]𝜑)
45 nfv 1883 . . . . . 6 𝑧𝑥𝐴 𝜑
46 nfcv 2793 . . . . . . 7 𝑦𝐴
4746, 2nfrex 3036 . . . . . 6 𝑦𝑥𝐴 [𝑧 / 𝑦]𝜑
483rexbidv 3081 . . . . . 6 (𝑦 = 𝑧 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑))
4945, 47, 48cbvral 3197 . . . . 5 (∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑 ↔ ∀𝑧 ∈ ran 𝑓𝑥𝐴 [𝑧 / 𝑦]𝜑)
5044, 49sylibr 224 . . . 4 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)
51 sseq1 3659 . . . . . 6 (𝑐 = ran 𝑓 → (𝑐𝐵 ↔ ran 𝑓𝐵))
52 rexeq 3169 . . . . . . 7 (𝑐 = ran 𝑓 → (∃𝑦𝑐 𝜑 ↔ ∃𝑦 ∈ ran 𝑓𝜑))
5352ralbidv 3015 . . . . . 6 (𝑐 = ran 𝑓 → (∀𝑥𝐴𝑦𝑐 𝜑 ↔ ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑))
54 raleq 3168 . . . . . 6 (𝑐 = ran 𝑓 → (∀𝑦𝑐𝑥𝐴 𝜑 ↔ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑))
5551, 53, 543anbi123d 1439 . . . . 5 (𝑐 = ran 𝑓 → ((𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑) ↔ (ran 𝑓𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)))
5655rspcev 3340 . . . 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 1368 . . 3 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∃𝑐 ∈ Fin (𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑))
588, 57exlimddv 1903 . 2 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑐 ∈ Fin (𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑))
59583adant2 1100 1 ((𝐴 ∈ Fin ∧ 𝐵𝑀 ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑐 ∈ Fin (𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  [wsbc 3468  wss 3607  ran crn 5144   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  Fincfn 7997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-fin 8001
This theorem is referenced by:  filbcmb  33665
  Copyright terms: Public domain W3C validator