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Mirrors > Home > MPE Home > Th. List > fsuppcolem | Structured version Visualization version GIF version |
Description: Lemma for fsuppco 8307. Formula building theorem for finite supports: rearranging the index set. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
fsuppcolem.f | ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin) |
fsuppcolem.g | ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌) |
Ref | Expression |
---|---|
fsuppcolem | ⊢ (𝜑 → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvco 5308 | . . . 4 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹) | |
2 | 1 | imaeq1i 5463 | . . 3 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) |
3 | imaco 5640 | . . 3 ⊢ ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) | |
4 | 2, 3 | eqtri 2644 | . 2 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) |
5 | fsuppcolem.g | . . . 4 ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌) | |
6 | df-f1 5893 | . . . . 5 ⊢ (𝐺:𝑋–1-1→𝑌 ↔ (𝐺:𝑋⟶𝑌 ∧ Fun ◡𝐺)) | |
7 | 6 | simprbi 480 | . . . 4 ⊢ (𝐺:𝑋–1-1→𝑌 → Fun ◡𝐺) |
8 | 5, 7 | syl 17 | . . 3 ⊢ (𝜑 → Fun ◡𝐺) |
9 | fsuppcolem.f | . . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin) | |
10 | imafi 8259 | . . 3 ⊢ ((Fun ◡𝐺 ∧ (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin) → (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin) | |
11 | 8, 9, 10 | syl2anc 693 | . 2 ⊢ (𝜑 → (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin) |
12 | 4, 11 | syl5eqel 2705 | 1 ⊢ (𝜑 → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 {csn 4177 ◡ccnv 5113 “ cima 5117 ∘ ccom 5118 Fun wfun 5882 ⟶wf 5884 –1-1→wf1 5885 Fincfn 7955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-fin 7959 |
This theorem is referenced by: fsuppco 8307 |
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