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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 9384, 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 6079 | . . 3 ⊢ (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ⊆ dom (𝑝 ∈ 𝑁 ↦ 𝑋) | |
3 | eqid 2727 | . . . 4 ⊢ (𝑝 ∈ 𝑁 ↦ 𝑋) = (𝑝 ∈ 𝑁 ↦ 𝑋) | |
4 | 3 | dmmptss 6239 | . . 3 ⊢ dom (𝑝 ∈ 𝑁 ↦ 𝑋) ⊆ 𝑁 |
5 | 2, 4 | sstri 3987 | . 2 ⊢ (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ⊆ 𝑁 |
6 | ssfi 9187 | . 2 ⊢ ((𝑁 ∈ Fin ∧ (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ⊆ 𝑁) → (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ∈ Fin) | |
7 | 1, 5, 6 | sylancl 585 | 1 ⊢ (𝜑 → (◡(𝑝 ∈ 𝑁 ↦ 𝑋) “ 𝑌) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ⊆ wss 3944 ↦ cmpt 5225 ◡ccnv 5671 dom cdm 5672 “ cima 5675 Fincfn 8953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7863 df-1o 8478 df-en 8954 df-fin 8957 |
This theorem is referenced by: (None) |
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