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Mirrors > Home > MPE Home > Th. List > cnvimamptfin | Structured version Visualization version GIF version |
Description: A preimage of a mapping with a finite domain under any class is finite. In contrast to fisuppfi 8993, the range of the mapping needs not to be known. (Contributed by AV, 21-Dec-2018.) |
Ref | Expression |
---|---|
cnvimamptfin.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
Ref | Expression |
---|---|
cnvimamptfin | ⊢ (𝜑 → (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvimamptfin.n | . 2 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
2 | cnvimass 5949 | . . 3 ⊢ (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ⊆ dom (𝑝 ∈ 𝑁 ↦ 𝑋) | |
3 | eqid 2737 | . . . 4 ⊢ (𝑝 ∈ 𝑁 ↦ 𝑋) = (𝑝 ∈ 𝑁 ↦ 𝑋) | |
4 | 3 | dmmptss 6104 | . . 3 ⊢ dom (𝑝 ∈ 𝑁 ↦ 𝑋) ⊆ 𝑁 |
5 | 2, 4 | sstri 3910 | . 2 ⊢ (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ⊆ 𝑁 |
6 | ssfi 8851 | . 2 ⊢ ((𝑁 ∈ Fin ∧ (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ⊆ 𝑁) → (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ∈ Fin) | |
7 | 1, 5, 6 | sylancl 589 | 1 ⊢ (𝜑 → (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ⊆ wss 3866 ↦ cmpt 5135 ◡ccnv 5550 dom cdm 5551 “ cima 5554 Fincfn 8626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-om 7645 df-1o 8202 df-en 8627 df-fin 8630 |
This theorem is referenced by: (None) |
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