cnvfiALT - Metamath Proof Explorer

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Theorem cnvfiALT 9267

Description: Shorter proof of cnvfi 9118 using ax-pow 5315. (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 6156	. . 3 ⊢ ^◡^◡𝐴 ⊆ 𝐴
2		ssfi 9115	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ^◡^◡𝐴 ⊆ 𝐴) → ^◡^◡𝐴 ∈ Fin)
3	1, 2	mpan2 691	. 2 ⊢ (𝐴 ∈ Fin → ^◡^◡𝐴 ∈ Fin)
4		relcnv 6065	. . 3 ⊢ Rel ^◡𝐴
5		cnvexg 7881	. . 3 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ∈ V)
6		cnven 8982	. . 3 ⊢ ((Rel ^◡𝐴 ∧ ^◡𝐴 ∈ V) → ^◡𝐴 ≈ ^◡^◡𝐴)
7	4, 5, 6	sylancr 587	. 2 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ≈ ^◡^◡𝐴)
8		enfii 9128	. 2 ⊢ ((^◡^◡𝐴 ∈ Fin ∧ ^◡𝐴 ≈ ^◡^◡𝐴) → ^◡𝐴 ∈ Fin)
9	3, 7, 8	syl2anc 584	1 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2109 Vcvv 3444 ⊆ wss 3911 class class class wbr 5102 ^◡ccnv 5630 Rel wrel 5636 ≈ cen 8893 Fincfn 8896

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7692

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6453 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-om 7824 df-1st 7948 df-2nd 7949 df-1o 8412 df-en 8897 df-fin 8900

This theorem is referenced by: (None)

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