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Mirrors > Home > MPE Home > Th. List > fsuppcolem | Structured version Visualization version GIF version |
Description: Lemma for fsuppco 9399. 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 5879 | . . . 4 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹) | |
2 | 1 | imaeq1i 6050 | . . 3 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) |
3 | imaco 6244 | . . 3 ⊢ ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) | |
4 | 2, 3 | eqtri 2754 | . 2 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) |
5 | fsuppcolem.g | . . . 4 ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌) | |
6 | df-f1 6542 | . . . . 5 ⊢ (𝐺:𝑋–1-1→𝑌 ↔ (𝐺:𝑋⟶𝑌 ∧ Fun ◡𝐺)) | |
7 | 6 | simprbi 496 | . . . 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 9177 | . . 3 ⊢ ((Fun ◡𝐺 ∧ (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin) → (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin) | |
11 | 8, 9, 10 | syl2anc 583 | . 2 ⊢ (𝜑 → (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin) |
12 | 4, 11 | eqeltrid 2831 | 1 ⊢ (𝜑 → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3468 ∖ cdif 3940 {csn 4623 ◡ccnv 5668 “ cima 5672 ∘ ccom 5673 Fun wfun 6531 ⟶wf 6533 –1-1→wf1 6534 Fincfn 8941 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-om 7853 df-1o 8467 df-en 8942 df-fin 8945 |
This theorem is referenced by: fsuppco 9399 |
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