cnvfiALT - Metamath Proof Explorer

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Theorem cnvfiALT 9309

Description: Shorter proof of cnvfi 9173 using ax-pow 5334. (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 6191	. . 3 ⊢ ^◡^◡𝐴 ⊆ 𝐴
2		ssfi 9170	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ^◡^◡𝐴 ⊆ 𝐴) → ^◡^◡𝐴 ∈ Fin)
3	1, 2	mpan2 704	. 2 ⊢ (𝐴 ∈ Fin → ^◡^◡𝐴 ∈ Fin)
4		relcnv 6104	. . 3 ⊢ Rel ^◡𝐴
5		cnvexg 7924	. . 3 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ∈ V)
6		cnven 9043	. . 3 ⊢ ((Rel ^◡𝐴 ∧ ^◡𝐴 ∈ V) → ^◡𝐴 ≈ ^◡^◡𝐴)
7	4, 5, 6	sylancr 599	. 2 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ≈ ^◡^◡𝐴)
8		enfii 9183	. 2 ⊢ ((^◡^◡𝐴 ∈ Fin ∧ ^◡𝐴 ≈ ^◡^◡𝐴) → ^◡𝐴 ∈ Fin)
9	3, 7, 8	syl2anc 596	1 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ∈ Fin)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∈ wcel 2145 Vcvv 3453 ⊆ wss 3902 class class class wbr 5107 ^◡ccnv 5658 Rel wrel 5664 ≈ cen 8952 Fincfn 8955

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-om 7866 df-1st 7989 df-2nd 7990 df-1o 8458 df-en 8956 df-fin 8959

This theorem is used by: (None)

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