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Mirrors > Home > MPE Home > Th. List > fsuppcolem | Structured version Visualization version GIF version |
Description: Lemma for fsuppco 9420. 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 5883 | . . . 4 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹) | |
2 | 1 | imaeq1i 6056 | . . 3 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) |
3 | imaco 6251 | . . 3 ⊢ ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) | |
4 | 2, 3 | eqtri 2753 | . 2 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) |
5 | fsuppcolem.g | . . . 4 ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌) | |
6 | df-f1 6548 | . . . . 5 ⊢ (𝐺:𝑋–1-1→𝑌 ↔ (𝐺:𝑋⟶𝑌 ∧ Fun ◡𝐺)) | |
7 | 6 | simprbi 495 | . . . 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 9193 | . . 3 ⊢ ((Fun ◡𝐺 ∧ (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin) → (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin) | |
11 | 8, 9, 10 | syl2anc 582 | . 2 ⊢ (𝜑 → (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin) |
12 | 4, 11 | eqeltrid 2829 | 1 ⊢ (𝜑 → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3463 ∖ cdif 3938 {csn 4625 ◡ccnv 5672 “ cima 5676 ∘ ccom 5677 Fun wfun 6537 ⟶wf 6539 –1-1→wf1 6540 Fincfn 8957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7866 df-1o 8480 df-en 8958 df-fin 8961 |
This theorem is referenced by: fsuppco 9420 |
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