Theorem cnvimamptfin 9263

Description: A preimage of a mapping with a finite domain under any class is finite. In contrast to fisuppfi 9284, the range of the mapping needs not to be known. (Contributed by AV, 21-Dec-2018.)

Hypothesis

Ref	Expression
cnvimamptfin.n	⊢ (𝜑 → 𝑁 ∈ Fin)

Assertion

Ref	Expression
cnvimamptfin	⊢ (𝜑 → (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ∈ Fin)

Distinct variable group:   𝑁,𝑝

Allowed substitution hints:   𝜑(𝑝)   𝑋(𝑝)   𝑌(𝑝)

Proof of Theorem cnvimamptfin

Step	Hyp	Ref	Expression
1		cnvimamptfin.n	. 2 ⊢ (𝜑 → 𝑁 ∈ Fin)
2		cnvimass 6047	. . 3 ⊢ (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ⊆ dom (𝑝 ∈ 𝑁 ↦ 𝑋)
3		eqid 2736	. . . 4 ⊢ (𝑝 ∈ 𝑁 ↦ 𝑋) = (𝑝 ∈ 𝑁 ↦ 𝑋)
4	3	dmmptss 6205	. . 3 ⊢ dom (𝑝 ∈ 𝑁 ↦ 𝑋) ⊆ 𝑁
5	2, 4	sstri 3931	. 2 ⊢ (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ⊆ 𝑁
6		ssfi 9107	. 2 ⊢ ((𝑁 ∈ Fin ∧ (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ⊆ 𝑁) → (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ∈ Fin)
7	1, 5, 6	sylancl 587	1 ⊢ (𝜑 → (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ∈ Fin)

Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3889   ↦ cmpt 5166  ^◡ccnv 5630  dom cdm 5631   “ cima 5634  Fincfn 8893

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 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  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 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-om 7818  df-1o 8405  df-en 8894  df-fin 8897

This theorem is referenced by:  elrgspnsubrunlem2  33309