cnvfiALT - Metamath Proof Explorer

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Theorem cnvfiALT 9353

Description: Shorter proof of cnvfi 9216 using ax-pow 5331. (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 6197	. . 3 ⊢ ^◡^◡𝐴 ⊆ 𝐴
2		ssfi 9213	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ^◡^◡𝐴 ⊆ 𝐴) → ^◡^◡𝐴 ∈ Fin)
3	1, 2	mpan2 704	. 2 ⊢ (𝐴 ∈ Fin → ^◡^◡𝐴 ∈ Fin)
4		relcnv 6111	. . 3 ⊢ Rel ^◡𝐴
5		cnvexg 7957	. . 3 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ∈ V)
6		cnven 9086	. . 3 ⊢ ((Rel ^◡𝐴 ∧ ^◡𝐴 ∈ V) → ^◡𝐴 ≈ ^◡^◡𝐴)
7	4, 5, 6	sylancr 599	. 2 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ≈ ^◡^◡𝐴)
8		enfii 9226	. 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 2147 Vcvv 3454 ⊆ wss 3900 class class class wbr 5106 ^◡ccnv 5654 Rel wrel 5660 ≈ cen 8995 Fincfn 8998

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-om 7899 df-1st 8022 df-2nd 8023 df-1o 8501 df-en 8999 df-fin 9002

This theorem is used by: (None)

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