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Mirrors > Home > MPE Home > Th. List > fsuppcolem | Structured version Visualization version GIF version |
Description: Lemma for fsuppco 8348. 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 5340 | . . . 4 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹) | |
2 | 1 | imaeq1i 5498 | . . 3 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) |
3 | imaco 5678 | . . 3 ⊢ ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) | |
4 | 2, 3 | eqtri 2673 | . 2 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) |
5 | fsuppcolem.g | . . . 4 ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌) | |
6 | df-f1 5931 | . . . . 5 ⊢ (𝐺:𝑋–1-1→𝑌 ↔ (𝐺:𝑋⟶𝑌 ∧ Fun ◡𝐺)) | |
7 | 6 | simprbi 479 | . . . 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 8300 | . . 3 ⊢ ((Fun ◡𝐺 ∧ (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin) → (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin) | |
11 | 8, 9, 10 | syl2anc 694 | . 2 ⊢ (𝜑 → (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin) |
12 | 4, 11 | syl5eqel 2734 | 1 ⊢ (𝜑 → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2030 Vcvv 3231 ∖ cdif 3604 {csn 4210 ◡ccnv 5142 “ cima 5146 ∘ ccom 5147 Fun wfun 5920 ⟶wf 5922 –1-1→wf1 5923 Fincfn 7997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-om 7108 df-1o 7605 df-er 7787 df-en 7998 df-dom 7999 df-fin 8001 |
This theorem is referenced by: fsuppco 8348 |
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