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Theorem fsuppcolem 8898
 Description: Lemma for fsuppco 8899. Formula building theorem for finite supports: rearranging the index set. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Hypotheses
Ref Expression
fsuppcolem.f (𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ Fin)
fsuppcolem.g (𝜑𝐺:𝑋1-1𝑌)
Assertion
Ref Expression
fsuppcolem (𝜑 → ((𝐹𝐺) “ (V ∖ {𝑍})) ∈ Fin)

Proof of Theorem fsuppcolem
StepHypRef Expression
1 cnvco 5725 . . . 4 (𝐹𝐺) = (𝐺𝐹)
21imaeq1i 5898 . . 3 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍}))
3 imaco 6081 . . 3 ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
42, 3eqtri 2781 . 2 ((𝐹𝐺) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
5 fsuppcolem.g . . . 4 (𝜑𝐺:𝑋1-1𝑌)
6 df-f1 6340 . . . . 5 (𝐺:𝑋1-1𝑌 ↔ (𝐺:𝑋𝑌 ∧ Fun 𝐺))
76simprbi 500 . . . 4 (𝐺:𝑋1-1𝑌 → Fun 𝐺)
85, 7syl 17 . . 3 (𝜑 → Fun 𝐺)
9 fsuppcolem.f . . 3 (𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ Fin)
10 imafi 8743 . . 3 ((Fun 𝐺 ∧ (𝐹 “ (V ∖ {𝑍})) ∈ Fin) → (𝐺 “ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
118, 9, 10syl2anc 587 . 2 (𝜑 → (𝐺 “ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
124, 11eqeltrid 2856 1 (𝜑 → ((𝐹𝐺) “ (V ∖ {𝑍})) ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  Vcvv 3409   ∖ cdif 3855  {csn 4522  ◡ccnv 5523   “ cima 5527   ∘ ccom 5528  Fun wfun 6329  ⟶wf 6331  –1-1→wf1 6332  Fincfn 8527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-om 7580  df-1o 8112  df-en 8528  df-fin 8531 This theorem is referenced by:  fsuppco  8899
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