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Theorem fsuppcolem 8347
Description: Lemma for fsuppco 8348. 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 5340 . . . 4 (𝐹𝐺) = (𝐺𝐹)
21imaeq1i 5498 . . 3 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍}))
3 imaco 5678 . . 3 ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
42, 3eqtri 2673 . 2 ((𝐹𝐺) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
5 fsuppcolem.g . . . 4 (𝜑𝐺:𝑋1-1𝑌)
6 df-f1 5931 . . . . 5 (𝐺:𝑋1-1𝑌 ↔ (𝐺:𝑋𝑌 ∧ Fun 𝐺))
76simprbi 479 . . . 4 (𝐺:𝑋1-1𝑌 → Fun 𝐺)
85, 7syl 17 . . 3 (𝜑 → Fun 𝐺)
9 fsuppcolem.f . . 3 (𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ Fin)
10 imafi 8300 . . 3 ((Fun 𝐺 ∧ (𝐹 “ (V ∖ {𝑍})) ∈ Fin) → (𝐺 “ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
118, 9, 10syl2anc 694 . 2 (𝜑 → (𝐺 “ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
124, 11syl5eqel 2734 1 (𝜑 → ((𝐹𝐺) “ (V ∖ {𝑍})) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  Vcvv 3231  cdif 3604  {csn 4210  ccnv 5142  cima 5146  ccom 5147  Fun wfun 5920  wf 5922  1-1wf1 5923  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-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:  fsuppco  8348
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