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Theorem isinffi 10030
Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 9294 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
Assertion
Ref Expression
isinffi ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵1-1𝐴)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem isinffi
Dummy variables 𝑐 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ficardom 9999 . . 3 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
2 isinf 9294 . . 3 𝐴 ∈ Fin → ∀𝑎 ∈ ω ∃𝑐(𝑐𝐴𝑐𝑎))
3 breq2 5152 . . . . . 6 (𝑎 = (card‘𝐵) → (𝑐𝑎𝑐 ≈ (card‘𝐵)))
43anbi2d 630 . . . . 5 (𝑎 = (card‘𝐵) → ((𝑐𝐴𝑐𝑎) ↔ (𝑐𝐴𝑐 ≈ (card‘𝐵))))
54exbidv 1919 . . . 4 (𝑎 = (card‘𝐵) → (∃𝑐(𝑐𝐴𝑐𝑎) ↔ ∃𝑐(𝑐𝐴𝑐 ≈ (card‘𝐵))))
65rspcva 3620 . . 3 (((card‘𝐵) ∈ ω ∧ ∀𝑎 ∈ ω ∃𝑐(𝑐𝐴𝑐𝑎)) → ∃𝑐(𝑐𝐴𝑐 ≈ (card‘𝐵)))
71, 2, 6syl2anr 597 . 2 ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑐(𝑐𝐴𝑐 ≈ (card‘𝐵)))
8 simprr 773 . . . . . 6 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → 𝑐 ≈ (card‘𝐵))
9 ficardid 10000 . . . . . . 7 (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
109ad2antlr 727 . . . . . 6 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → (card‘𝐵) ≈ 𝐵)
11 entr 9045 . . . . . 6 ((𝑐 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝑐𝐵)
128, 10, 11syl2anc 584 . . . . 5 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → 𝑐𝐵)
1312ensymd 9044 . . . 4 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → 𝐵𝑐)
14 bren 8994 . . . 4 (𝐵𝑐 ↔ ∃𝑓 𝑓:𝐵1-1-onto𝑐)
1513, 14sylib 218 . . 3 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → ∃𝑓 𝑓:𝐵1-1-onto𝑐)
16 f1of1 6848 . . . . . 6 (𝑓:𝐵1-1-onto𝑐𝑓:𝐵1-1𝑐)
17 simplrl 777 . . . . . 6 ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵1-1-onto𝑐) → 𝑐𝐴)
18 f1ss 6810 . . . . . 6 ((𝑓:𝐵1-1𝑐𝑐𝐴) → 𝑓:𝐵1-1𝐴)
1916, 17, 18syl2an2 686 . . . . 5 ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵1-1-onto𝑐) → 𝑓:𝐵1-1𝐴)
2019ex 412 . . . 4 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → (𝑓:𝐵1-1-onto𝑐𝑓:𝐵1-1𝐴))
2120eximdv 1915 . . 3 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → (∃𝑓 𝑓:𝐵1-1-onto𝑐 → ∃𝑓 𝑓:𝐵1-1𝐴))
2215, 21mpd 15 . 2 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → ∃𝑓 𝑓:𝐵1-1𝐴)
237, 22exlimddv 1933 1 ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  wral 3059  wss 3963   class class class wbr 5148  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  ωcom 7887  cen 8981  Fincfn 8984  cardccrd 9973
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-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
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-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-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-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-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-om 7888  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977
This theorem is referenced by:  fidomtri  10031  hashdom  14415  erdsze2lem1  35188  eldioph2lem2  42749
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