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Theorem indexfi 8439
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 33842. (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 1992 . . . . . 6 𝑧𝜑
2 nfsbc1v 3596 . . . . . 6 𝑦[𝑧 / 𝑦]𝜑
3 sbceq1a 3587 . . . . . 6 (𝑦 = 𝑧 → (𝜑[𝑧 / 𝑦]𝜑))
41, 2, 3cbvrex 3307 . . . . 5 (∃𝑦𝐵 𝜑 ↔ ∃𝑧𝐵 [𝑧 / 𝑦]𝜑)
54ralbii 3118 . . . 4 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑧𝐵 [𝑧 / 𝑦]𝜑)
6 dfsbcq 3578 . . . . 5 (𝑧 = (𝑓𝑥) → ([𝑧 / 𝑦]𝜑[(𝑓𝑥) / 𝑦]𝜑))
76ac6sfi 8369 . . . 4 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑧𝐵 [𝑧 / 𝑦]𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑))
85, 7sylan2b 493 . . 3 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑))
9 simpll 807 . . . . 5 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → 𝐴 ∈ Fin)
10 ffn 6206 . . . . . . 7 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
1110ad2antrl 766 . . . . . 6 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → 𝑓 Fn 𝐴)
12 dffn4 6282 . . . . . 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 8417 . . . . 5 ((𝐴 ∈ Fin ∧ 𝑓:𝐴onto→ran 𝑓) → ran 𝑓 ∈ Fin)
159, 13, 14syl2anc 696 . . . 4 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ran 𝑓 ∈ Fin)
16 frn 6214 . . . . 5 (𝑓:𝐴𝐵 → ran 𝑓𝐵)
1716ad2antrl 766 . . . 4 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ran 𝑓𝐵)
18 fnfvelrn 6519 . . . . . . . . 9 ((𝑓 Fn 𝐴𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
1910, 18sylan 489 . . . . . . . 8 ((𝑓:𝐴𝐵𝑥𝐴) → (𝑓𝑥) ∈ ran 𝑓)
20 rspesbca 3661 . . . . . . . . 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 3100 . . . . . 6 (𝑓:𝐴𝐵 → (∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑 → ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑))
2423imp 444 . . . . 5 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑) → ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑)
2524adantl 473 . . . 4 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑)
26 simpr 479 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) ∧ 𝑤𝐴) → 𝑤𝐴)
27 simprr 813 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)
28 nfv 1992 . . . . . . . . . . 11 𝑤[(𝑓𝑥) / 𝑦]𝜑
29 nfsbc1v 3596 . . . . . . . . . . 11 𝑥[𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑
30 fveq2 6352 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝑓𝑥) = (𝑓𝑤))
3130sbceq1d 3581 . . . . . . . . . . . 12 (𝑥 = 𝑤 → ([(𝑓𝑥) / 𝑦]𝜑[(𝑓𝑤) / 𝑦]𝜑))
32 sbceq1a 3587 . . . . . . . . . . . 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 3306 . . . . . . . . . 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 3070 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) ∧ 𝑤𝐴) → [𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑)
37 rspesbca 3661 . . . . . . . 8 ((𝑤𝐴[𝑤 / 𝑥][(𝑓𝑤) / 𝑦]𝜑) → ∃𝑥𝐴 [(𝑓𝑤) / 𝑦]𝜑)
3826, 36, 37syl2anc 696 . . . . . . 7 ((((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) ∧ 𝑤𝐴) → ∃𝑥𝐴 [(𝑓𝑤) / 𝑦]𝜑)
3938ralrimiva 3104 . . . . . 6 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∀𝑤𝐴𝑥𝐴 [(𝑓𝑤) / 𝑦]𝜑)
40 dfsbcq 3578 . . . . . . . . 9 (𝑧 = (𝑓𝑤) → ([𝑧 / 𝑦]𝜑[(𝑓𝑤) / 𝑦]𝜑))
4140rexbidv 3190 . . . . . . . 8 (𝑧 = (𝑓𝑤) → (∃𝑥𝐴 [𝑧 / 𝑦]𝜑 ↔ ∃𝑥𝐴 [(𝑓𝑤) / 𝑦]𝜑))
4241ralrn 6525 . . . . . . 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 1992 . . . . . 6 𝑧𝑥𝐴 𝜑
46 nfcv 2902 . . . . . . 7 𝑦𝐴
4746, 2nfrex 3145 . . . . . 6 𝑦𝑥𝐴 [𝑧 / 𝑦]𝜑
483rexbidv 3190 . . . . . 6 (𝑦 = 𝑧 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑))
4945, 47, 48cbvral 3306 . . . . 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 3767 . . . . . 6 (𝑐 = ran 𝑓 → (𝑐𝐵 ↔ ran 𝑓𝐵))
52 rexeq 3278 . . . . . . 7 (𝑐 = ran 𝑓 → (∃𝑦𝑐 𝜑 ↔ ∃𝑦 ∈ ran 𝑓𝜑))
5352ralbidv 3124 . . . . . 6 (𝑐 = ran 𝑓 → (∀𝑥𝐴𝑦𝑐 𝜑 ↔ ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑))
54 raleq 3277 . . . . . 6 (𝑐 = ran 𝑓 → (∀𝑦𝑐𝑥𝐴 𝜑 ↔ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑))
5551, 53, 543anbi123d 1548 . . . . 5 (𝑐 = ran 𝑓 → ((𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑) ↔ (ran 𝑓𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓𝑥𝐴 𝜑)))
5655rspcev 3449 . . . 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 1479 . . 3 (((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑓𝑥) / 𝑦]𝜑)) → ∃𝑐 ∈ Fin (𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑))
588, 57exlimddv 2012 . 2 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑐 ∈ Fin (𝑐𝐵 ∧ ∀𝑥𝐴𝑦𝑐 𝜑 ∧ ∀𝑦𝑐𝑥𝐴 𝜑))
59583adant2 1126 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 1072   = wceq 1632  wex 1853  wcel 2139  wral 3050  wrex 3051  [wsbc 3576  wss 3715  ran crn 5267   Fn wfn 6044  wf 6045  ontowfo 6047  cfv 6049  Fincfn 8121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-1o 7729  df-er 7911  df-en 8122  df-dom 8123  df-fin 8125
This theorem is referenced by:  filbcmb  33848
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