cnvfiALT - Metamath Proof Explorer

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Theorem cnvfiALT 9299

Description: Shorter proof of cnvfi 9163 using ax-pow 5340. (Contributed by Mario Carneiro, 28-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

**Assertion**
Ref	Expression
cnvfiALT	⊢ (𝐴 ∈ Fin → ^◡𝐴 ∈ Fin)

**Proof of Theorem cnvfiALT**
Step	Hyp	Ref	Expression
1		cnvcnvss 6196	. . 3 ⊢ ^◡^◡𝐴 ⊆ 𝐴
2		ssfi 9160	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ^◡^◡𝐴 ⊆ 𝐴) → ^◡^◡𝐴 ∈ Fin)
3	1, 2	mpan2 703	. 2 ⊢ (𝐴 ∈ Fin → ^◡^◡𝐴 ∈ Fin)
4		relcnv 6110	. . 3 ⊢ Rel ^◡𝐴
5		cnvexg 7924	. . 3 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ∈ V)
6		cnven 9033	. . 3 ⊢ ((Rel ^◡𝐴 ∧ ^◡𝐴 ∈ V) → ^◡𝐴 ≈ ^◡^◡𝐴)
7	4, 5, 6	sylancr 598	. 2 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ≈ ^◡^◡𝐴)
8		enfii 9173	. 2 ⊢ ((^◡^◡𝐴 ∈ Fin ∧ ^◡𝐴 ≈ ^◡^◡𝐴) → ^◡𝐴 ∈ Fin)
9	3, 7, 8	syl2anc 595	1 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2150 Vcvv 3462 ⊆ wss 3913 class class class wbr 5114 ^◡ccnv 5664 Rel wrel 5670 ≈ cen 8943 Fincfn 8946

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2152 ax-9 2160 ax-10 2183 ax-11 2199 ax-12 2220 ax-ext 2742 ax-sep 5262 ax-nul 5274 ax-pow 5340 ax-pr 5408 ax-un 7736

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2099 df-mo 2574 df-eu 2604 df-clab 2749 df-cleq 2762 df-clel 2845 df-nfc 2919 df-ne 2966 df-ral 3087 df-rex 3097 df-reu 3377 df-rab 3424 df-v 3464 df-sbc 3753 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5560 df-eprel 5565 df-po 5573 df-so 5574 df-fr 5618 df-we 5620 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-om 7866 df-1st 7989 df-2nd 7990 df-1o 8456 df-en 8947 df-fin 8950

This theorem is referenced by: (None)

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