cnvfiALT - Metamath Proof Explorer

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Theorem cnvfiALT 9241

Description: Shorter proof of cnvfi 9102 using ax-pow 5309. (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 6151	. . 3 ⊢ ^◡^◡𝐴 ⊆ 𝐴
2		ssfi 9099	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ^◡^◡𝐴 ⊆ 𝐴) → ^◡^◡𝐴 ∈ Fin)
3	1, 2	mpan2 692	. 2 ⊢ (𝐴 ∈ Fin → ^◡^◡𝐴 ∈ Fin)
4		relcnv 6062	. . 3 ⊢ Rel ^◡𝐴
5		cnvexg 7866	. . 3 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ∈ V)
6		cnven 8972	. . 3 ⊢ ((Rel ^◡𝐴 ∧ ^◡𝐴 ∈ V) → ^◡𝐴 ≈ ^◡^◡𝐴)
7	4, 5, 6	sylancr 588	. 2 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ≈ ^◡^◡𝐴)
8		enfii 9112	. 2 ⊢ ((^◡^◡𝐴 ∈ Fin ∧ ^◡𝐴 ≈ ^◡^◡𝐴) → ^◡𝐴 ∈ Fin)
9	3, 7, 8	syl2anc 585	1 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3439 ⊆ wss 3900 class class class wbr 5097 ^◡ccnv 5622 Rel wrel 5628 ≈ cen 8882 Fincfn 8885

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-om 7809 df-1st 7933 df-2nd 7934 df-1o 8397 df-en 8886 df-fin 8889

This theorem is referenced by: (None)

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