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Theorem fsuppcolem 8251
Description: Lemma for fsuppco 8252. 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 5273 . . . 4 (𝐹𝐺) = (𝐺𝐹)
21imaeq1i 5426 . . 3 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍}))
3 imaco 5602 . . 3 ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
42, 3eqtri 2648 . 2 ((𝐹𝐺) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
5 fsuppcolem.g . . . 4 (𝜑𝐺:𝑋1-1𝑌)
6 df-f1 5855 . . . . 5 (𝐺:𝑋1-1𝑌 ↔ (𝐺:𝑋𝑌 ∧ Fun 𝐺))
76simprbi 480 . . . 4 (𝐺:𝑋1-1𝑌 → Fun 𝐺)
85, 7syl 17 . . 3 (𝜑 → Fun 𝐺)
9 fsuppcolem.f . . 3 (𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ Fin)
10 imafi 8204 . . 3 ((Fun 𝐺 ∧ (𝐹 “ (V ∖ {𝑍})) ∈ Fin) → (𝐺 “ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
118, 9, 10syl2anc 692 . 2 (𝜑 → (𝐺 “ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
124, 11syl5eqel 2708 1 (𝜑 → ((𝐹𝐺) “ (V ∖ {𝑍})) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1992  Vcvv 3191  cdif 3557  {csn 4153  ccnv 5078  cima 5082  ccom 5083  Fun wfun 5844  wf 5846  1-1wf1 5847  Fincfn 7900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-1o 7506  df-er 7688  df-en 7901  df-dom 7902  df-fin 7904
This theorem is referenced by:  fsuppco  8252
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